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Volume 8, Issue 1

# On Dirichlet problem for fractional p-Laplacian with singular non-linearity

Tuhina Mukherjee
• Corresponding author
• Department of Mathematics, Indian Institute of Technology Delhi Hauz Khaz, New Delhi -16, India
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Published Online: 2016-12-02 | DOI: https://doi.org/10.1515/anona-2016-0100

## Abstract

In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity:

where Ω is a bounded domain in ${ℝ}^{n}$ with smooth boundary $\partial \mathrm{\Omega }$, $n>sp$, $s\in \left(0,1\right)$, $\lambda >0$, $0 and $1. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

MSC 2010: 35R11; 35R09; 35A15

## 1 Introduction

Let $s\in \left(0,1\right)$, $p\in \left(1,\mathrm{\infty }\right)$ and let $\mathrm{\Omega }\subset {ℝ}^{n}$, $n\ge 2$, be a bounded domain with smooth boundary. We consider the following problem with singular non-linearity:

(Pλ)

where $\lambda >0$, $0, $p<\alpha +1\le {p}_{s}^{*}$, ${p}_{s}^{*}=\frac{np}{n-sp}$, $n>sp$ and ${\left(-{\mathrm{\Delta }}_{p}\right)}^{s}$ is the fractional p-Laplacian operator defined as

Recently, a lot of attention is given to the study of fractional and non-local operators of elliptic type due to concrete real-world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc.

The semilinear Dirichlet problem for the fractional Laplacian is recently studied, using variational methods, in [8, 42, 43]. The existence and multiplicity results for non-local operators, like the fractional Laplacian with a combination of convex and concave type non-linearities like ${u}^{q}+\lambda {u}^{p}$, p, $q>0$, is studied in [3, 5, 37, 36, 44, 45]. The eigenvalue problem for the fractional p-Laplacian and properties like the simplicity of the smallest eigenvalue is studied in [16, 32]. The Brezis–Nirenberg type existence result has been studied in [38]. Existence and multiplicity results with convex-concave type regular non-linearities have been studied in [26].

In the local setting ($s=1$), the paper by Crandal, Rabinowitz and Tartar [12] is the starting point on semilinear problems with singular non-linearity. A lot of work has been done related to the existence and multiplicity results for the Laplacian and the p-Laplacian with singular non-linearity, see [1, 11, 14, 20, 18, 23, 24]. In [11, 14], the singular problems of the type

was considered with $g\left(x,u\right)\sim {u}^{-\alpha }$. The existence of solutions under suitable conditions on g and h was studied. In [20, 18], Ghergu and Rădulescu considered singular problems of the type

where λ, $\mu >0$. Here, $h,K\in {C}^{0,\gamma }\left(\mathrm{\Omega }\right)$ for some $0<\gamma <1$ and $h>0$ in Ω, and $f:\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a Hölder continuous function which is positive on $\overline{\mathrm{\Omega }}×\left(0,\mathrm{\infty }\right)$, sublinear at $\mathrm{\infty }$ and superlinear at 0. The function $g\in {C}^{0,\gamma }\left(0,\mathrm{\infty }\right)$, for some $0<\gamma <1$, is non negative and non-increasing function such that ${lim}_{s\to {0}^{+}}g\left(s\right)=+\mathrm{\infty }$. They proved several results related to existence and non existence of positive solutions of the above problems, by taking into account both the sign of the potential K and the decay rate around the origin of the singular non-linearity g. Several authors considered problems of Lane–Emden–Fowler type with singular non-linearity, see [10, 15, 19]. In addition, some bifurcation results have been proved in [19] for the following problem:

where $\lambda ,\mu >0$, $0, f is non-decreasing with respect to the second variable and $g\left(u\right)$ behaves like ${u}^{-\alpha }$ around the origin. The asymptotic behaviour of the solutions is shown by constructing suitable sub- and super-solutions combined with the maximum principle. We also refer to [22, 29] as a part of previous contributions to this field. For detailed study and recent results on singular problems, we refer to [21].

In [23], Giacomoni, Schindler and Taḱaĉ studied the critical growth singular problem

where $0<\delta <1$ and $p-1, ${p}^{*}=\frac{np}{n-p}$ and ${\mathrm{\Delta }}_{p}u=\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right)$. Using variational methods, they proved the existence of multiple solutions with restrictions on p and q in the spirit of [13, 17]. Among the works dealing with elliptic equations with singular and critical growth terms, we cite [1, 34, 2, 9, 25, 40], see also the references therein, with no attempt to provide a complete list.

There are many works on the existence of a solution for fractional elliptic equations with regular non-linearities like ${u}^{q}+\lambda {u}^{p}$, $p,q>0$. Sub-critical growth problems are studied in [8, 42, 43], and critical exponent problems are studied in [5, 37, 36, 38]. Also, the multiplicity of solutions by the method of Nehari manifold and fibering maps has been investigated in [26, 45, 46]. For detailed study and recent results on this subject, we refer to [35].

In [4], Barrios et al. studied the singular problem

where $n>2s$, $M\ge 0$, $0, $\gamma ,\lambda >0$, $1, ${2}_{s}^{*}=\frac{2n}{n-2s}$, $f\in {L}^{m}\left(\mathrm{\Omega }\right)$ is a non-negative function for $m\ge 1$, and

In particular, they studied the existence of distributional solutions for small λ using the uniform estimates of $\left\{{u}_{n}\right\}$ which are solutions of the regularised problem with the singular term ${u}^{-\gamma }$ replaced by ${\left(u+\frac{1}{n}\right)}^{-\gamma }$. In [39], the critical growth singular problem, for $\alpha ={2}_{s}^{*}-1$ and $p=2$, is studied where multiplicity results are obtained using the Nehari manifold approach.

There are many works on the study of p-fractional equations with polynomial type non-linearities. In [26], Goyal and the second author studied the subcritical problem using the Nehari manifold and fibering maps. In [38], a Brezis–Nirenberg type critical exponent problem is studied. We also refer readers to [6, 27, 31] and the references therein. To the best of our knowledge, there are no works on the existence of multiplicity results with singular non-linearities involving the fractional p-Laplacian operator.

In this paper, we study existence and multiplicity results with a convex-concave type singular non-linearity. Here we follow the approach of [30]. We obtain our results by studying the existence of minimizers that arise out of the structure of the Nehari manifold. We would like to remark that the results proved here are new even for the case $q=1$. Also the existence result is sharp in the sense that we show the existence of Λ such that $\left(0,\mathrm{\Lambda }\right)$ is the maximal range for λ for which the solution exists. We show the existence of a second solution in the sub-critical case for the maximal range of λ, where the fibering maps have two critical points along with some other condition. We also show some regularity results on the weak solutions of ().

The paper is organised as follows. In Section 2, we present some preliminaries on function spaces required for the variational setting and state the main results. In Section 3, we study the corresponding Nehari manifold and the properties of minimizers. In Sections 4 and 5, we show the existence of minimizers and solutions. In Section 6, we show some regularity results, and Section 7 is devoted to the maximal range of λ for the existence of solutions.

## 2 Preliminaries and main results

The motivation for defining the function space comes from [42]. In [26], Goyal and the second author discussed the Dirichlet boundary value problem involving the p-fractional Laplace operator using variational techniques. Due to non-localness of the fractional p-Laplace operator, they introduced the function space $\left({X}_{0},\parallel \cdot {\parallel }_{{X}_{0}}\right)$. The space X is defined as

where $Q={ℝ}^{2n}\setminus \left(\mathcal{𝒞}\mathrm{\Omega }×\mathcal{𝒞}\mathrm{\Omega }\right)$ and $\mathcal{𝒞}\mathrm{\Omega }:={ℝ}^{n}\setminus \mathrm{\Omega }$. The space X is endowed with the norm

Then we define . There exists a constant $C>0$ such that ${\parallel u\parallel }_{{L}^{p}\left(\mathrm{\Omega }\right)}\le C{\left[u\right]}_{X}$ for all $u\in {X}_{0}$. Hence, $\parallel u\parallel ={\left[u\right]}_{X}$ is a norm on ${X}_{0}$, and ${X}_{0}$ is a Hilbert space. Note that the norm $\parallel \cdot \parallel$ involves the interaction between Ω and ${ℝ}^{n}\setminus \mathrm{\Omega }$. We denote $\parallel \cdot {\parallel }_{{L}^{p}\left(\mathrm{\Omega }\right)}$ as $|\cdot {|}_{p}$ and $\parallel \cdot \parallel ={\left[\cdot \right]}_{X}$ for the norm in ${X}_{0}$. Now, for each $\beta \ge 0$, we set

${C}_{\beta }=sup\left\{{|u|}_{\beta }^{\beta }:u\in {X}_{0},\parallel u\parallel =1\right\}.$(2.1)

Then ${C}_{0}=|\mathrm{\Omega }|$ = Lebesgue measure of Ω and ${\int }_{\mathrm{\Omega }}|u{|}^{\beta }dx\le {C}_{\beta }\parallel u{\parallel }^{\beta }$ for all $u\in {X}_{0}$. From the embedding results in [26], we know that ${X}_{0}$ is continuously and compactly embedded in ${L}^{r}\left(\mathrm{\Omega }\right)$, where $1\le r<{p}_{s}^{*}$ and the embedding is continuous but not compact if $r={p}_{s}^{*}$. We define the best constant S of the embedding as

$S=inf\left\{{\parallel u\parallel }^{p}:u\in {X}_{0},{|u|}_{{p}_{s}^{*}}^{p}=1\right\}.$

#### Definition 2.1.

We say that $u\in {X}_{0}$ is a positive weak solution of () if $u>0$ in Ω and

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}-{u}^{\alpha }\right)\psi 𝑑x=0$

for all $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

The functional associated to (), ${I}_{\lambda }:{X}_{0}\to \left(-\mathrm{\infty },\mathrm{\infty }\right]$, is defined by

${I}_{\lambda }\left(u\right)=\frac{1}{p}{\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}dxdy-\lambda {\int }_{\mathrm{\Omega }}{G}_{q}\left(u\right)dx-\frac{1}{\alpha +1}{\int }_{\mathrm{\Omega }}|u{|}^{\alpha +1}dx,$

where ${G}_{q}:ℝ\to \left[-\mathrm{\infty },\mathrm{\infty }\right)$ is the function defined by

For each $0, we set ${X}_{+}=\left\{u\in {X}_{0}:u\ge 0\right\}$ and ${X}_{+,q}=\left\{u\in {X}_{+}:u\not\equiv 0,{G}_{q}\left(u\right)\in {L}^{1}\left(\mathrm{\Omega }\right)\right\}$. Notice that ${X}_{+,q}={X}_{+}\setminus \left\{0\right\}$ if $0 and ${X}_{+,1}\ne \mathrm{\varnothing }$ if $\partial \mathrm{\Omega }$ is, for example, of ${C}^{2}$. We will need the following important lemma.

#### Lemma 2.2.

For each $w\mathrm{\in }{X}_{\mathrm{+}}$, there exists a sequence $\mathrm{\left\{}{w}_{k}\mathrm{\right\}}$ in ${X}_{\mathrm{0}}$ such that ${w}_{k}\mathrm{\to }w$ strongly in ${X}_{\mathrm{0}}$, where $\mathrm{0}\mathrm{\le }{w}_{\mathrm{1}}\mathrm{\le }{w}_{\mathrm{2}}\mathrm{\le }\mathrm{\cdots }$ and ${w}_{k}$ has compact support in Ω for each k.

#### Proof.

The proof here is adopted from [30]. Let $w\in {X}_{+}$ and let $\left\{{\psi }_{k}\right\}$ be a sequence in ${C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ such that ${\psi }_{k}$ is non negative and converges strongly to w in ${X}_{0}$. Define ${z}_{k}=\mathrm{min}\left\{{\psi }_{k},w\right\}$. Then ${z}_{k}\to w$ strongly in ${X}_{0}$. Now, we set ${w}_{1}={z}_{{r}_{1}}$ where ${r}_{1}>0$ is such that $\parallel {z}_{{r}_{1}}-w\parallel \le 1$. Then $\mathrm{max}\left\{{w}_{1},{z}_{m}\right\}\to w$ strongly in ${X}_{0}$, thus we can find ${r}_{2}>0$ such that $\parallel \mathrm{max}\left\{{w}_{1},{z}_{{r}_{2}}\right\}-w\parallel \le \frac{1}{2}$. We set ${w}_{2}=\mathrm{max}\left\{{w}_{1},{z}_{{r}_{2}}\right\}$, and get $\mathrm{max}\left\{{w}_{2},{z}_{m}\right\}\to w$ strongly as $m\to \mathrm{\infty }$. Consequently, by induction, we set ${w}_{k+1}=\mathrm{max}\left\{{w}_{k},{z}_{{r}_{k+1}}\right\}$ to obtain the desired sequence, since we can see that ${w}_{k}\in {X}_{0}$ has compact support for each k and $\parallel \mathrm{max}\left\{{w}_{k},{z}_{{r}_{k+1}}\right\}-w\parallel \le \frac{1}{k+1}$, which says that $\left\{{w}_{k}\right\}$ converges strongly to w in ${X}_{0}$. ∎

Let ${\varphi }_{1}>0$ be the eigenfunction of ${\left(-{\mathrm{\Delta }}_{p}\right)}^{s}$ corresponding to the smallest eigenvalue ${\lambda }_{1}$. This is obtained as minimizer of the minimization problem

${\lambda }_{1}=\mathrm{min}\left\{{\parallel u\parallel }^{p}:u\in {X}_{0},{|u|}_{p}=1\right\}.$

In [32, 38], it was shown that this minimizer is achieved by a unique positive and bounded function ${\varphi }_{1}$. Moreover, $\left({\lambda }_{1},{\varphi }_{1}\right)$ is the solution of the eigenvalue problem

We assume ${\parallel {\varphi }_{1}\parallel }_{{L}^{\mathrm{\infty }}}=1$. With these preliminaries, we state our main results.

For each $u\in {X}_{+,q}$, we define the fiber map ${\varphi }_{u}:{ℝ}^{+}\to ℝ$ by ${\varphi }_{u}\left(t\right)={I}_{\lambda }\left(tu\right)$.

#### Theorem 2.3.

Assume $\mathrm{0}\mathrm{<}q\mathrm{\le }\mathrm{1}$. In the case $q\mathrm{=}\mathrm{1}$, assume also ${X}_{\mathrm{+}\mathrm{,}\mathrm{1}}\mathrm{\ne }\mathrm{\varnothing }$. Let ${\mathrm{\Lambda }}_{\mathrm{1}}$ be a constant defined by

Then ${\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{>}\mathrm{0}$.

#### Theorem 2.4.

For all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{\right)}$, () has at least two distinct solutions in ${X}_{\mathrm{+}\mathrm{,}q}$ when $\alpha \mathrm{<}{p}_{s}^{\mathrm{*}}\mathrm{-}\mathrm{1}$ and at least one solution in the critical case when $\alpha \mathrm{=}{p}_{s}^{\mathrm{*}}\mathrm{-}\mathrm{1}$.

#### Definition 2.5.

We say that $u\in {X}_{0}$ is a weak sub-solution of () if $u>0$ in Ω and

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y\le {\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right)\psi 𝑑x=0$

for all $0\le \psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. Similarly, $u\in {X}_{0}$ is said to be a weak super-solution to () if in the above, the reverse inequalities hold.

Next we study that the existence of a solution with the parameter in the maximal interval. For this, we minimize the functional over the convex set $\left\{u\in {X}_{+,q}:\underset{¯}{u}\le u\le \overline{u}\right\}$, where $\underset{¯}{u}$ and $\overline{u}$ are sub- and super-solutions, respectively. Using truncation techniques as in [28], we show that the minimizer is a weak solution.

#### Theorem 2.6.

Let $p\mathrm{<}\alpha \mathrm{+}\mathrm{1}\mathrm{\le }{p}_{s}^{\mathrm{*}}$ and $\mathrm{0}\mathrm{<}q\mathrm{\le }\mathrm{1}$. Then there exists $\mathrm{\Lambda }\mathrm{>}\mathrm{0}$ such that () has a solution for all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{\Lambda }\mathrm{\right)}$ and no solution for $\lambda \mathrm{>}\mathrm{\Lambda }$.

## 3 Nehari manifold and fibering maps

We denote ${I}_{\lambda }=I$ for simplicity now. One can easily verify that the energy functional I is not bounded below on the space ${X}_{0}$. We will show that it is bounded on the manifold associated to the functional I. In this section, we study the structure of this manifold. We define

${\mathcal{𝒩}}_{\lambda }=\left\{u\in {X}_{+,q}:〈{I}^{\prime }\left(u\right),u〉=0\right\}.$

#### Theorem 3.1.

The functional I is coercive and bounded below on ${\mathcal{N}}_{\lambda }$.

#### Proof.

In the case $0, since $u\in {\mathcal{𝒩}}_{\lambda }$, using the embedding of ${X}_{0}$ in ${L}^{1-q}\left(\mathrm{\Omega }\right)$, we get

$I\left(u\right)=\left(\frac{1}{p}-\frac{1}{\alpha +1}\right)\parallel u{\parallel }^{p}-\lambda \left(\frac{1}{1-q}-\frac{1}{\alpha +1}\right){\int }_{\mathrm{\Omega }}|u{|}^{1-q}dx\ge {c}_{1}\parallel u{\parallel }^{p}-{c}_{2}\parallel u{\parallel }^{1-q}$

for some constants ${c}_{1}$ and ${c}_{2}$. This says that I is coercive and bounded below on ${\mathcal{𝒩}}_{\lambda }$.

In the case $q=1$, using the inequality $\mathrm{ln}\left(|u|\right)\le |u|$ and embedding results for ${X}_{0}$, we can similarly get I as bounded below. ∎

From the definition of fiber map ${\varphi }_{u}$, we have

which gives

${\varphi }_{u}^{\prime }\left(t\right)={t}^{p-1}\parallel u{\parallel }^{p}-\lambda {t}^{-q}{\int }_{\mathrm{\Omega }}|u{|}^{1-q}dx-{t}^{\alpha }{\int }_{\mathrm{\Omega }}|u{|}^{\alpha +1}dx,$${\varphi }_{u}^{\prime \prime }\left(t\right)=\left(p-1\right){t}^{p-2}\parallel u{\parallel }^{p}+q\lambda {t}^{-q-1}{\int }_{\mathrm{\Omega }}|u{|}^{1-q}dx-\alpha {t}^{\alpha -1}{\int }_{\mathrm{\Omega }}|u{|}^{\alpha +1}dx.$

It is easy to see that the points in ${\mathcal{𝒩}}_{\lambda }$ are corresponding to the critical points of ${\varphi }_{u}$ at $t=1$. So, it is natural to divide ${\mathcal{𝒩}}_{\lambda }$ into three sets corresponding to local minima, local maxima and points of inflexion. Therefore, we define

${\mathcal{𝒩}}_{\lambda }^{+}=\left\{u\in {\mathcal{𝒩}}_{\lambda }:{\varphi }_{u}^{\prime }\left(1\right)=0,{\varphi }_{u}^{\prime \prime }\left(1\right)>0\right\}=\left\{{t}_{0}u\in {\mathcal{𝒩}}_{\lambda }:{t}_{0}>0,{\varphi }_{u}^{\prime }\left({t}_{0}\right)=0,{\varphi }_{u}^{\prime \prime }\left({t}_{0}\right)>0\right\},$${\mathcal{𝒩}}_{\lambda }^{-}=\left\{u\in {\mathcal{𝒩}}_{\lambda }:{\varphi }_{u}^{\prime }\left(1\right)=0,{\varphi }_{u}^{\prime \prime }\left(1\right)<0\right\}=\left\{{t}_{0}u\in {\mathcal{𝒩}}_{\lambda }:{t}_{0}>0,{\varphi }_{u}^{\prime }\left({t}_{0}\right)=0,{\varphi }_{u}^{\prime \prime }\left({t}_{0}\right)<0\right\}$

and ${\mathcal{𝒩}}_{\lambda }^{0}=\left\{u\in {\mathcal{𝒩}}_{\lambda }:{\varphi }_{u}^{\prime }\left(1\right)=0,{\varphi }_{u}^{\prime \prime }\left(1\right)=0\right\}$.

#### Lemma 3.2.

There exists ${\lambda }_{\mathrm{*}}\mathrm{>}\mathrm{0}$ such that for each $u\mathrm{\in }{X}_{\mathrm{+}\mathrm{,}q}\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$, there are unique ${t}_{\mathrm{1}}$ and ${t}_{\mathrm{2}}$ with the property that ${t}_{\mathrm{1}}\mathrm{<}{t}_{\mathrm{2}}$, ${t}_{\mathrm{1}}\mathit{}u\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and ${t}_{\mathrm{2}}\mathit{}u\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{-}}$ for all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\lambda }_{\mathrm{*}}\mathrm{\right)}$.

#### Proof.

Define $A\left(u\right)={\int }_{\mathrm{\Omega }}|u{|}^{1-q}dx$ and $B\left(u\right)={\int }_{\mathrm{\Omega }}|u{|}^{\alpha +1}dx$. Let $u\in {X}_{+,q}$. Then we have

$\frac{d}{dt}I\left(tu\right)={t}^{p-1}{\parallel u\parallel }^{p}-{t}^{-q}\lambda A\left(u\right)-{t}^{\alpha }B\left(u\right)={t}^{-q}\left({m}_{u}\left(t\right)-\lambda A\left(u\right)\right),$

and we define ${m}_{u}\left(t\right):={t}^{p-1+q}{\parallel u\parallel }^{p}-{t}^{\alpha +q}B\left(u\right)$. Since ${lim}_{t\to \mathrm{\infty }}{m}_{u}\left(t\right)=-\mathrm{\infty }$, we can easily see that ${m}_{u}\left(t\right)$ attains its maximum at

${t}_{\mathrm{max}}={\left[\frac{\left(p-1+q\right){\parallel u\parallel }^{p}}{\left(\alpha +q\right)B\left(u\right)}\right]}^{\frac{1}{\alpha +1-p}}$

and

${m}_{u}\left({t}_{\mathrm{max}}\right)=\left(\frac{\alpha +2-p}{p-1+q}\right){\left(\frac{p-1+q}{\alpha +q}\right)}^{\frac{\alpha +q}{\alpha +1-p}}\frac{{\parallel u\parallel }^{\frac{p\left(\alpha +q\right)}{\alpha +1-p}}}{B{\left(u\right)}^{\frac{p-1+q}{\alpha +1-p}}}.$

Now, $u\in {\mathcal{𝒩}}_{\lambda }$ if and only if ${m}_{u}\left(t\right)=\lambda A\left(u\right)$, and we see that

${m}_{u}\left(t\right)-\lambda A\left(u\right)\ge {m}_{u}\left({t}_{\mathrm{max}}\right)-\lambda {|u|}_{1-q}^{1-q}\ge \left(\frac{\alpha +2-p}{p-1+q}\right){\left(\frac{p-1+q}{\alpha +q}\right)}^{\frac{\alpha +q}{\alpha +1-p}}\frac{{\parallel u\parallel }^{\frac{p\left(\alpha +q\right)}{\alpha +1-p}}}{B{\left(u\right)}^{\frac{p-1+q}{\alpha +1-p}}}-\lambda {C}_{1-q}{\parallel u\parallel }^{1-q}>0$

if and only if

$\lambda <\mathit{=:}\left(\frac{\alpha +2-p}{p-1+q}\right){\left(\frac{p-1+q}{\alpha +q}\right)}^{\frac{\alpha +q}{\alpha +1-p}}{\left({C}_{\alpha +1}\right)}^{\frac{-p+1-q}{\alpha +1-p}}{C}_{1-q}^{-1}{\lambda }_{*},$

where ${C}_{\alpha +1}$ and ${C}_{1-q}$ are defined in (2.1).

In the case $0, we see that ${m}_{u}\left(t\right)=\lambda {\int }_{\mathrm{\Omega }}|u{|}^{1-q}dx$ if and only if ${\varphi }_{u}^{\prime }\left(t\right)=0$. So for $\lambda \in \left(0,{\lambda }_{*}\right)$, there exist exactly two points $0<{t}_{1}<{t}_{\mathrm{max}}<{t}_{2}$ with ${m}_{u}^{\prime }\left({t}_{1}\right)>0$ and ${m}_{u}^{\prime }\left({t}_{2}\right)<0$, that is, ${t}_{1}u\in {\mathcal{𝒩}}_{\lambda }^{+}$ and ${t}_{2}u\in {\mathcal{𝒩}}_{\lambda }^{-}$. Thus, ${\varphi }_{u}$ has a local minimum at $t={t}_{1}$ and a local maximum at $t={t}_{2}$. In other words, ${\varphi }_{u}$ is decreasing in $\left(0,{t}_{1}\right)$ and increasing in $\left({t}_{1},{t}_{2}\right)$.

In the case $q=1$, since ${lim}_{t\to 0}{\varphi }_{u}\left(t\right)=\mathrm{\infty }$ and ${lim}_{t\to \mathrm{\infty }}{\varphi }_{u}\left(t\right)=-\mathrm{\infty }$, with similar reasoning as above, we get ${t}_{1},{t}_{2}$.

In both cases, ${\varphi }_{u}$ has exactly two critical points ${t}_{1}$ and ${t}_{2}$ such that $0<{t}_{1}<{t}_{\mathrm{max}}<{t}_{2}$, ${\varphi }_{u}^{\prime \prime }\left({t}_{1}\right)>0$ and ${\varphi }_{u}^{\prime \prime }\left({t}_{2}\right)<0$. Thus, ${t}_{1}u\in {\mathcal{𝒩}}_{\lambda }^{+}$, ${t}_{2}u\in {\mathcal{𝒩}}_{\lambda }^{-}$. ∎

#### Proof of Theorem 2.3.

The proof follows from Lemma 3.2. ∎

#### Corollary 3.3.

We have ${\mathcal{N}}_{\lambda }^{\mathrm{0}}\mathrm{=}\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$ for all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{\right)}$.

#### Proof.

Let $u\in {\mathcal{𝒩}}_{\lambda }^{0}$ and $u\not\equiv 0$. Then $u\in {\mathcal{𝒩}}_{\lambda }$, that is, $t=1$ is a critical point of ${\varphi }_{u}\left(t\right)$. By Lemma 3.2, ${\varphi }_{u}$ has critical points corresponding to either local minima or local maxima. So, $t=1$ is the critical point corresponding to either local minima or local maxima of ${\varphi }_{u}$. Thus, either $u\in {\mathcal{𝒩}}_{\lambda }^{+}$ or $u\in {\mathcal{𝒩}}_{\lambda }^{-}$, which is a contradiction. ∎

We can now show that I is bounded below on ${\mathcal{𝒩}}_{\lambda }^{+}$ and ${\mathcal{𝒩}}_{\lambda }^{-}$ as follows.

#### Lemma 3.4.

The following hold:

• (i)

$sup\left\{\parallel u\parallel :u\in {\mathcal{𝒩}}_{\lambda }^{+}\right\}<\mathrm{\infty }$,

• (ii)

$inf\left\{\parallel v\parallel :v\in {\mathcal{𝒩}}_{\lambda }^{-}\right\}>0$ and $sup\left\{\parallel v\parallel :v\in {\mathcal{𝒩}}_{\lambda }^{-},I\left(v\right)\le M\right\}<\mathrm{\infty }$ for each $M>0$.

Moreover, $\mathrm{inf}\mathit{}I\mathit{}\mathrm{\left(}{\mathcal{N}}_{\lambda }^{\mathrm{+}}\mathrm{\right)}\mathrm{>}\mathrm{-}\mathrm{\infty }$ and $\mathrm{inf}\mathit{}I\mathit{}\mathrm{\left(}{\mathcal{N}}_{\lambda }^{\mathrm{-}}\mathrm{\right)}\mathrm{>}\mathrm{-}\mathrm{\infty }$.

#### Proof.

Let $u\in {\mathcal{𝒩}}_{\lambda }^{+}$ and $v\in {\mathcal{𝒩}}_{\lambda }^{-}$. Then we have

$0<{\varphi }_{u}^{\prime \prime }\left(1\right)\le \left(p-1-\alpha \right){\parallel u\parallel }^{p}+\lambda \left(\alpha +q\right){C}_{1-q}{\parallel u\parallel }^{1-q},$$0>{\varphi }_{v}^{\prime \prime }\left(1\right)\ge \left(p-1+q\right){\parallel v\parallel }^{p}-\left(\alpha +q\right){C}_{\alpha +1}{\parallel v\parallel }^{\alpha +1}.$

Thus, we obtain

$\parallel u\parallel \le {\left(\frac{\lambda \left(\alpha +q\right){C}_{1-q}}{\alpha +1-p}\right)}^{\frac{1}{p+q-1}}\mathit{ }\text{and}\mathit{ }\parallel v\parallel \ge {\left(\frac{p-1+q}{\left(\alpha +q\right){C}_{\alpha +1}}\right)}^{\frac{1}{\alpha +1-p}}.$

This implies that

$sup\left\{\parallel u\parallel :u\in {\mathcal{𝒩}}_{\lambda }^{+}\right\}<\mathrm{\infty }\mathit{ }\text{and}\mathit{ }inf\left\{\parallel v\parallel :v\in {\mathcal{𝒩}}_{\lambda }^{-}\right\}>0.$(3.1)

If $I\left(v\right)\le M$, using $\mathrm{ln}\left(|v|\right)\le |v|$, we get

(3.2)

This implies $sup\left\{\parallel v\parallel :v\in {\mathcal{𝒩}}_{\lambda }^{-},I\left(v\right)\le M\right\}<\mathrm{\infty }$ for each $M>0$. Using (3.1) and (3.2), it is easy to show that $infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)>-\mathrm{\infty }$ and $infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)>-\mathrm{\infty }$. ∎

#### Lemma 3.5.

Suppose that $u\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and $v\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{-}}$ are minimizers of I over ${\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and ${\mathcal{N}}_{\lambda }^{\mathrm{-}}$, respectively. Then, for each $w\mathrm{\in }{X}_{\mathrm{+}}$, the following hold:

• (1)

there exists ${ϵ}_{0}>0$ such that $I\left(u+ϵw\right)\ge I\left(u\right)$ for each $ϵ\in \left[0,{ϵ}_{0}\right]$,

• (2)

${t}_{ϵ}\to 1$ as $ϵ\to {0}^{+}$ , where ${t}_{ϵ}$ is the unique positive real number satisfying ${t}_{ϵ}\left(v+ϵw\right)\in {\mathcal{𝒩}}_{\lambda }^{-}$.

#### Proof.

(1) Let $w\in {X}_{+}$, i.e., $w\in {X}_{0}$ and $w\ge 0$. We set

$\rho \left(ϵ\right)=\left(p-1\right)\parallel u+ϵw{\parallel }^{p}+\lambda q{\int }_{\mathrm{\Omega }}|u+ϵw{|}^{1-q}dx-\alpha {\int }_{\mathrm{\Omega }}|u+ϵw{|}^{\alpha +1}dx$

for each $ϵ\ge 0$. Then, using the continuity of ρ, and since $u\in {\mathcal{𝒩}}_{\lambda }^{+}$, $\rho \left(0\right)={\varphi }_{u}^{\prime \prime }\left(1\right)>0$, there exists ${ϵ}_{0}>0$ such that $\rho \left(ϵ\right)>0$ for $ϵ\in \left[0,{ϵ}_{0}\right]$. Since, for each $ϵ>0$, there exists ${t}_{ϵ}^{\prime }>0$ such that ${t}_{ϵ}^{\prime }\left(u+ϵw\right)\in {\mathcal{𝒩}}_{\lambda }^{+}$, it follows that ${t}_{ϵ}^{\prime }\to 1$ as $ϵ\to 0$, and for each $ϵ\in \left[0,{ϵ}_{0}\right]$, we have

$I\left(u+ϵw\right)\ge I\left({t}_{ϵ}^{\prime }\left(u+ϵw\right)\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=I\left(u\right).$

(2) We define $h:\left(0,\mathrm{\infty }\right)×{ℝ}^{3}\to ℝ$ by

$h\left(t,{l}_{1},{l}_{2},{l}_{3}\right)={l}_{1}{t}^{p-1}-\lambda {t}^{-q}{l}_{2}-{t}^{\alpha }{l}_{3}$

for $\left(t,{l}_{1},{l}_{2},{l}_{3}\right)\in \left(0,\mathrm{\infty }\right)×{ℝ}^{3}$. Then h is a ${C}^{\mathrm{\infty }}$ function. Therefore, we have

$\frac{dh}{dt}\left(1,\parallel v{\parallel }^{p},{\int }_{\mathrm{\Omega }}|v{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v{|}^{\alpha +1}dx\right)={\varphi }_{v}^{\prime \prime }\left(1\right)<0,$

and for each $ϵ\ge 0$,

$h\left({t}_{ϵ},\parallel v+ϵw{\parallel }^{p},{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v{|}^{\alpha +1}dx\right)={\varphi }_{v+ϵw}^{\prime }\left({t}_{ϵ}\right)=0.$

Also,

$h\left(1,\parallel v{\parallel }^{p},{\int }_{\mathrm{\Omega }}|v{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v{|}^{\alpha +1}dx\right)={\varphi }_{v}^{\prime }\left(1\right)=0.$

Thus, by the implicit function theorem, there exists an open neighbourhood $A\subset \left(0,\mathrm{\infty }\right)$ and $B\subset {ℝ}^{3}$ containing 1 and $\left(\parallel v{\parallel }^{p},{\int }_{\mathrm{\Omega }}|v{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v{|}^{\alpha +1}\right)$, respectively, such that for all $y\in B$, $h\left(t,y\right)=0$ has a unique solution $t=g\left(y\right)\in A$, where $g:B\to A$ is a continuous function. So,

$\left(\parallel v+ϵw{\parallel }^{p},{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{\alpha +1}dx\right)\in B$

and

$g\left(\parallel v+ϵw\right){\parallel }^{p},{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{\alpha +1}dx\right)={t}_{ϵ},$

since

$h\left({t}_{ϵ},\parallel v+ϵw\right){\parallel }^{p},{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{1-q}dx,{\int }_{\mathrm{\Omega }}|v+ϵw{|}^{\alpha +1}dx\right)=0.$

Thus, by the continuity of g, we get ${t}_{ϵ}\to 1$ as $ϵ\to {0}^{+}$. ∎

#### Lemma 3.6.

Suppose that $u\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and $v\mathrm{\in }{N}_{\lambda }^{\mathrm{-}}$ are minimizers of I on ${\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and ${\mathcal{N}}_{\lambda }^{\mathrm{-}}$, respectively. Then, for each $w\mathrm{\in }{X}_{\mathrm{+}}$, we have ${u}^{\mathrm{-}q}\mathit{}w\mathrm{,}{v}^{\mathrm{-}q}\mathit{}w\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$,

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-\lambda {\int }_{\mathrm{\Omega }}\left({u}^{-q}+{u}^{\alpha }\right)w𝑑x\ge 0$(3.3)

and

${\int }_{Q}\frac{{|v\left(x\right)-v\left(y\right)|}^{p-2}\left(v\left(x\right)-v\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-\lambda {\int }_{\mathrm{\Omega }}\left({v}^{-q}+{v}^{\alpha }\right)w𝑑x\ge 0.$(3.4)

#### Proof.

Let $w\in {X}_{+}$. For sufficiently small $ϵ>0$, by Lemma 3.5,

$0\le \frac{I\left(u+ϵw\right)-I\left(u\right)}{ϵ}$$=\frac{1}{pϵ}\left({\parallel u+ϵw\parallel }^{p}-{\parallel u\parallel }^{p}\right)-\frac{\lambda }{ϵ}{\int }_{\mathrm{\Omega }}\left({G}_{q}\left(u+ϵw\right)-{G}_{q}\left(u\right)\right)𝑑x-\frac{1}{ϵ\left(\alpha +1\right)}{\int }_{\mathrm{\Omega }}\left({|u+ϵw|}^{\alpha +1}-{|u|}^{\alpha +1}\right)𝑑x.$(3.5)

We can easily verify that as $ϵ\to {0}^{+}$,

$\frac{\left({\parallel u+ϵw\parallel }^{p}-{\parallel u\parallel }^{p}\right)}{ϵ}\to p{\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y,$${\int }_{\mathrm{\Omega }}\frac{\left({|u+ϵw|}^{\alpha +1}-{|u|}^{\alpha +1}\right)}{ϵ}dx\to \left(\alpha +1\right){\int }_{\mathrm{\Omega }}|u{|}^{\alpha -1}uwdx,$

which imply that $\frac{\left({G}_{q}\left(u+ϵw\right)-{G}_{q}\left(u\right)\right)}{ϵ}\in {L}^{1}\left(\mathrm{\Omega }\right)$. Also, for each $x\in \mathrm{\Omega }$,

which increases monotonically as $ϵ↓0$ and

So, by using the monotone convergence theorem for the sequence $\left\{{G}_{q}\right\}$, we get ${u}^{-q}w\in {L}^{1}\left(\mathrm{\Omega }\right)$. Letting $ϵ↓0$ in both sides of (3.5), we get (3.3).

Next we will show these properties for v. For each $ϵ>0$, there exists ${t}_{ϵ}>0$ with ${t}_{ϵ}\left(v+ϵw\right)\in {\mathcal{𝒩}}_{\lambda }^{-}$. By Lemma 3.5 (2), for sufficiently small $ϵ>0$, we have

$I\left({t}_{ϵ}\left(v+ϵw\right)\right)\ge I\left(v\right)\ge I\left({t}_{ϵ}v\right),$

which implies $I\left({t}_{ϵ}\left(v+ϵw\right)\right)-I\left(v\right)\ge 0$. Thus, we have

$\lambda {\int }_{\mathrm{\Omega }}\left({G}_{q}\left({t}_{ϵ}{|v+ϵw|}^{1-q}\right)-{G}_{q}\left({|v|}^{1-q}\right)\right)𝑑x\le \frac{{t}_{ϵ}^{p}}{p}\left({\parallel v+ϵw\parallel }^{p}-{\parallel v\parallel }^{p}\right)-\frac{{t}_{ϵ}^{\alpha +1}}{\alpha +1}{\int }_{\mathrm{\Omega }}\left({|v+ϵw|}^{\alpha +1}-{|v|}^{\alpha +1}\right)𝑑x.$

As $ϵ↓0$, ${t}_{ϵ}\to 1$. Thus, using similar arguments as above, we obtain ${v}^{-q}w\in {L}^{1}\left(\mathrm{\Omega }\right)$ and (3.4) follows. ∎

Let $\eta >0$ be such that $\varphi =\eta {\varphi }_{1}$ satisfies

${\int }_{Q}\frac{{|\varphi \left(x\right)-\varphi \left(y\right)|}^{p-2}\left(\varphi \left(x\right)-\varphi \left(y\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y\le \lambda {\int }_{Q}{\varphi }^{-q}\psi 𝑑x+{\int }_{Q}{\varphi }^{\alpha }\psi 𝑑x$(3.6)

for all $0\le \psi \in {X}_{0}$ (i.e., ϕ is a sub-solution of ()) and ${\varphi }^{\alpha +q}\left(x\right)\le \lambda \left(\frac{q}{\alpha }\right)$ for each $x\in \mathrm{\Omega }$. Then we have the following lemma.

#### Lemma 3.7.

Suppose $u\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and $v\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{-}}$ are minimizers of I on ${\mathcal{N}}_{\lambda }^{\mathrm{+}}$ and ${\mathcal{N}}_{\lambda }^{\mathrm{-}}$, respectively. Then $u\mathrm{\ge }\varphi$ and $v\mathrm{\ge }\varphi$ in Ω. In particular, $u\mathrm{,}v\mathrm{>}\mathrm{0}$.

#### Proof.

By Lemma 2.2, let $\left\{{w}_{k}\right\}$ be a sequence in ${X}_{0}$ such that supp$\left({w}_{k}\right)$ is compact, $0\le {w}_{k}\le {\left(\varphi -u\right)}^{+}$ for each k and $\left\{{w}_{k}\right\}$ strongly converges to ${\left(\varphi -u\right)}^{+}$ in ${X}_{0}$. Then

$\frac{d}{dt}\left(\lambda {t}^{-q}+{t}^{\alpha }\right)=-q\lambda {t}^{-q-1}+\alpha {t}^{\alpha -1}\le 0⇔{t}^{\alpha +q}\le \lambda \left(\frac{q}{\alpha }\right).$(3.7)

Using Lemma 3.6 and (3.6), we have

${\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({w}_{k}\left(x\right)-{w}_{k}\left(y\right)\right)𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){w}_{k}𝑑x+{\int }_{\mathrm{\Omega }}\left(\lambda {\varphi }^{-q}+{\varphi }^{\alpha }\right){w}_{k}𝑑x\ge 0,$

where $f\left(\xi \right)={|\xi \left(x\right)-\xi \left(y\right)|}^{p-2}\left(\xi \left(x\right)-\xi \left(y\right)\right)$. Since $\left\{{w}_{k}\right\}$ converges to ${\left(\varphi -u\right)}^{+}$ strongly, we get a subsequence of $\left\{{w}_{k}\right\}$, still calling it $\left\{{w}_{k}\right\}$, such that ${w}_{k}\left(x\right)\to {\left(\varphi -u\right)}^{+}\left(x\right)$ pointwise almost everywhere in Ω, and we write for each $x\in \mathrm{\Omega }$, ${w}_{k}\left(x\right)={\left(\varphi -u\right)}^{+}\left(x\right)+o\left(1\right)$ as $k\to \mathrm{\infty }$. Then

${\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({w}_{k}\left(x\right)-{w}_{k}\left(y\right)\right)𝑑x𝑑y$$={\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y+o\left(1\right){\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y.$

Further, we can see that

${\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y$$=\left({\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}+{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}+{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{2}}\right)\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)dxdy,$(3.8)

where ${\mathrm{\Omega }}_{1}=\left\{x:\varphi \left(x\right)\ge u\left(x\right)\right\}$ and ${\mathrm{\Omega }}_{2}=\left\{x:\varphi \left(x\right)\le u\left(x\right)\right\}$. Now, we separately estimate each integrals and to begin with, first we see that

${\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{2}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y=0.$(3.9)

Next we see that

${\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y=-{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{\left(f\left(\varphi \right)-f\left(u\right)\right)}{{|x-y|}^{n+sp}}\left(\left(\varphi -u\right)\left(x\right)-\left(\varphi -u\right)\left(y\right)\right)𝑑x𝑑y$$\le -\frac{1}{{2}^{p-2}}{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|\left(\varphi -u\right)\left(x\right)-\left(\varphi -u\right)\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}𝑑x𝑑y$(3.10)

using ${|a-b|}^{p}\le {2}^{p-2}\left({|a|}^{p-2}a-{|b|}^{p-2}b\right)\left(a-b\right)$, $p\ge 2$ and $a,b\in ℝ$. Now, consider

${\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y$$={\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left(\varphi -u\right)\left(x\right)𝑑x𝑑y$$\le -\frac{1}{{2}^{p-2}}{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{{|\left(\varphi -u\right)\left(x\right)-\left(\varphi -u\right)\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}𝑑x𝑑y+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left(\varphi -u\right)\left(y\right)𝑑x𝑑y,$(3.11)

and, similarly, we get

${\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y$$\le -\frac{1}{{2}^{p-2}}{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{{|\left(\varphi -u\right)\left(x\right)-\left(\varphi -u\right)\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left(\varphi -u\right)\left(x\right)𝑑x𝑑y.$(3.12)

Thus, using (3.8)–(3.12), we get

${\int }_{Q}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left({\left(\varphi -u\right)}^{+}\left(x\right)-{\left(\varphi -u\right)}^{+}\left(y\right)\right)𝑑x𝑑y$$\le -\frac{1}{{2}^{p-2}}{\parallel \left(\varphi -u\right)\parallel }^{p}+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left(\varphi -u\right)\left(y\right)𝑑x𝑑y-{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{\left(f\left(u\right)-f\left(\varphi \right)\right)}{{|x-y|}^{n+sp}}\left(\varphi -u\right)\left(x\right)𝑑x𝑑y$$=-\frac{1}{{2}^{p-2}}{\parallel \left(\varphi -u\right)\parallel }^{p}.$

Since ${\varphi }^{\alpha +q}\left(x\right)\le \lambda \left(\frac{q}{\alpha }\right)$, for each $x\in \mathrm{\Omega }$, using (3.7), we get

${\int }_{\mathrm{\Omega }}\left(\left(\lambda {u}^{-q}+{u}^{\alpha }\right)-\left(\lambda {\varphi }^{-q}+{\varphi }^{\alpha }\right)\right){w}_{k}𝑑x={\int }_{\mathrm{\Omega }\cap \left\{\varphi \ge u\right\}}\left(\left(\lambda {u}^{-q}+{u}^{\alpha }\right)-\left(\lambda {\varphi }^{-q}+{\varphi }^{\alpha }\right)\right){\left(\varphi -u\right)}^{+}\left(x\right)𝑑x+o\left(1\right)\ge 0,$

which implies

$0\le -\frac{1}{{2}^{p-2}}{\parallel {\left(\varphi -u\right)}^{+}\parallel }^{p}-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){w}_{k}𝑑x+{\int }_{\mathrm{\Omega }}\left(\lambda {\varphi }^{-q}+{\varphi }^{\alpha }\right){w}_{k}𝑑x+o\left(1\right)\le -\frac{1}{{2}^{p-2}}{\parallel {\left(\varphi -u\right)}^{+}\parallel }^{p}+o\left(1\right),$

and letting $k\to \mathrm{\infty }$, we get $-{\parallel {\left(\varphi -u\right)}^{+}\parallel }^{2}\ge 0$. Thus, we showed $u\ge \varphi$. Similarly, we can show $v\ge \varphi$. ∎

## 4 Existence of minimizer on ${\mathcal{𝒩}}_{\lambda }^{+}$

In this section, we will show that the minimum of I on ${\mathcal{𝒩}}_{\lambda }^{+}$ is achieved in ${\mathcal{𝒩}}_{\lambda }^{+}$. Also, we show that this minimizer is also a solution of ().

#### Proposition 4.1.

For all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{\right)}$, there exist ${u}_{\lambda }\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{+}}$ satisfying $I\mathit{}\mathrm{\left(}{u}_{\lambda }\mathrm{\right)}\mathrm{=}\mathrm{inf}\mathit{}I\mathit{}\mathrm{\left(}{\mathcal{N}}_{\lambda }^{\mathrm{+}}\mathrm{\right)}$.

#### Proof.

Assume $0 and $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$. We show that there exists ${u}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{+}$ such that $I\left({u}_{\lambda }\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)$. Let $\left\{{u}_{k}\right\}\subset {\mathcal{𝒩}}_{\lambda }^{+}$ be a sequence such that $I\left({u}_{k}\right)\to infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)$ as $k\to \mathrm{\infty }$. Now, by (3.1), we can assume that there exists ${u}_{\lambda }\in {X}_{0}$ such that ${u}_{k}⇀{u}_{\lambda }$ weakly in ${X}_{0}$ (up to a subsequence). First we will show that $infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)<0$. Let ${u}_{0}\in {\mathcal{𝒩}}_{\lambda }^{+}$. We have ${\varphi }_{{u}_{0}}^{\prime \prime }\left(1\right)>0$, which gives

$\left(\frac{p-1+q}{\alpha +q}\right)\parallel {u}_{0}{\parallel }^{p}>{\int }_{\mathrm{\Omega }}|{u}_{0}{|}^{\alpha +1}dx.$

Therefore, using $\alpha +1>p$, we obtain

$I\left({u}_{0}\right)=\left(\frac{1}{p}-\frac{1}{1-q}\right)\parallel {u}_{0}{\parallel }^{p}+\left(\frac{1}{1-q}-\frac{1}{\alpha +1}\right){\int }_{\mathrm{\Omega }}|{u}_{0}{|}^{\alpha +1}dx$$\le -\frac{\left(p+q-1\right)}{p\left(1-q\right)}{\parallel {u}_{0}\parallel }^{p}+\frac{\left(p+q-1\right)}{\left(\alpha +1\right)\left(1-q\right)}{\parallel {u}_{0}\parallel }^{p}$$=\left(\frac{1}{\alpha +1}-\frac{1}{p}\right)\left(\frac{p+q-1}{1-q}\right){\parallel {u}_{0}\parallel }^{p}$$<0.$

This proves that $infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)<0$. Case (I): $\alpha \mathrm{+}\mathrm{1}\mathrm{<}{p}_{s}^{\mathrm{*}}$ and $\mathrm{0}\mathrm{<}q\mathrm{\le }\mathrm{1}$. First, we claim that ${u}_{\lambda }\in {X}_{+,q}$. When $0, if ${u}_{\lambda }=0$, then $0=I\left({u}_{\lambda }\right)\le {\underset{¯}{\mathrm{lim}}}_{k\to \mathrm{\infty }}I\left({u}_{k}\right)<0$, which is a contradiction. In the case $q=1$, the sequence $\left\{{\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{k}|\right)\right\}$ is bounded, since the sequence $\left\{I\left({u}_{k}\right)\right\}$ and $\left\{\parallel {u}_{k}\parallel \right\}$ is bounded. So, using Fatou’s lemma and the fact that $\mathrm{ln}\left(|{u}_{k}|\right)\le {u}_{k}$ for each k, we get

$-\mathrm{\infty }<{\overline{\mathrm{lim}}}_{k\to \mathrm{\infty }}{\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{k}|\right)𝑑x\le {\int }_{\mathrm{\Omega }}{\overline{\mathrm{lim}}}_{k\to \mathrm{\infty }}\mathrm{ln}\left(|{u}_{k}|\right)dx={\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{\lambda }|\right)𝑑x,$

which implies ${u}_{\lambda }\not\equiv 0$ and thus, in both cases we have shown ${u}_{\lambda }\in {X}_{+,q}$. We claim that ${u}_{k}\to {u}_{\lambda }$ strongly in ${X}_{0}$. Suppose not. Then, we may assume $\parallel {u}_{k}-{u}_{\lambda }\parallel \to c>0$. Using the Brezis–Lieb lemma and the embedding results for ${X}_{0}$ in the subcritical case, we have

$\underset{k\to \mathrm{\infty }}{lim}{\varphi }_{{u}_{k}}^{\prime }\left(1\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left(1\right)+{c}^{p},$(4.1)

which implies ${\varphi }_{{u}_{\lambda }}^{\prime }\left(1\right)+{c}^{p}=0$, using the fact that ${\varphi }_{{u}_{k}}^{\prime }\left(1\right)=0$ for each k. Since $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$, there exist $0<{t}_{1}<{t}_{2}$ (by fibering map analysis) such that ${\varphi }_{{u}_{\lambda }}^{\prime }\left({t}_{1}\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left({t}_{2}\right)=0$ and ${t}_{1}{u}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{+}$. By (4.1), we have ${\varphi }_{{u}_{\lambda }}^{\prime }\left(1\right)<0$, which gives two cases: $1<{t}_{1}$ or ${t}_{2}<1$. When ${t}_{1}>1$, we have

$infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=limI\left({u}_{k}\right)=I\left({u}_{\lambda }\right)+\frac{{c}^{p}}{p}={\varphi }_{{u}_{\lambda }}\left(1\right)+\frac{{c}^{p}}{p}>{\varphi }_{{u}_{\lambda }}\left(1\right)>{\varphi }_{{u}_{\lambda }}\left({t}_{1}\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right),$

which is a contradiction. Thus, we have ${t}_{2}<1$. We set $f\left(t\right)={\varphi }_{{u}_{\lambda }}\left(t\right)+\frac{{c}^{p}{t}^{p}}{2}$ for $t>0$. From (4.1), we get ${f}^{\prime }\left(1\right)=0$, and since $0<{t}_{2}<1$, ${f}^{\prime }\left({t}_{2}\right)={t}_{2}^{p-1}{c}^{p}>0$. So, f is increasing in $\left[{t}_{2},1\right]$, and we obtain

$infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=I\left({u}_{\lambda }\right)+\frac{{c}^{p}}{p}={\varphi }_{{u}_{\lambda }}\left(1\right)+\frac{{c}^{p}}{p}=f\left(1\right)>f\left({t}_{2}\right)>{\varphi }_{{u}_{\lambda }}\left({t}_{2}\right)>{\varphi }_{{u}_{\lambda }}\left({t}_{1}\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right),$

which gives a contradiction. Hence, $c=0$, and thus ${u}_{k}\to {u}_{\lambda }$ strongly in ${X}_{0}$. Since $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$, we have ${\varphi }_{{u}_{\lambda }}^{\prime \prime }\left(1\right)>0$, so we obtain ${u}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{+}$ and $I\left({u}_{\lambda }\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)$. Case (II): $\alpha \mathrm{=}{p}_{s}^{\mathrm{*}}\mathrm{-}\mathrm{1}$ and $\mathrm{0}\mathrm{<}q\mathrm{<}\mathrm{1}$. We set ${w}_{k}:={u}_{k}-{u}_{\lambda }$ and claim that ${u}_{k}\to {u}_{\lambda }$ strongly in ${X}_{0}$. Suppose ${\parallel {w}_{k}\parallel }^{p}\to {c}^{p}\ne 0$ and ${\int }_{\mathrm{\Omega }}|{w}_{k}{|}^{{p}_{s}^{*}}dx\to {d}^{{p}_{s}^{*}}$ as $k\to \mathrm{\infty }$. Since ${u}_{k}\in {\mathcal{𝒩}}_{\lambda }^{+}$, using the Brezis–Lieb lemma, we get

$0=\underset{k\to \mathrm{\infty }}{lim}{\varphi }_{{u}_{k}}^{\prime }\left(1\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left(1\right)+{c}^{p}-{d}^{{p}_{s}^{*}},$(4.2)

which implies

$\parallel {u}_{\lambda }{\parallel }^{p}+{c}^{p}=\lambda {\int }_{\mathrm{\Omega }}|{u}_{\lambda }{|}^{1-q}dx+{\int }_{\mathrm{\Omega }}|{u}_{k}{|}^{{p}_{s}^{*}}dx+{d}^{{p}_{s}^{*}}.$

We claim that ${u}_{\lambda }\in {X}_{+,q}$. Suppose ${u}_{\lambda }\equiv 0$. If $0 and $c=0$, then $0>infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=I\left(0\right)=0$, which is a contradiction, and if $c\ne 0$, then

$infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=I\left(0\right)+\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}=\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}.$(4.3)

But we have $S{\parallel {u}_{k}\parallel }_{{p}_{s}^{*}}^{p}\le {\parallel {u}_{k}\parallel }^{p}$, which gives ${c}^{p}\ge S{d}^{p}$. Also from (4.2) we have ${c}^{p}={d}^{{p}_{s}^{*}}$. Then (4.3) implies

$0>infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=\left(\frac{1}{p}-\frac{1}{{p}_{s}^{*}}\right){c}^{p}\ge \frac{s}{n}{S}^{\frac{n}{sp}},$

which is again a contradiction. In the case $q=1$, the sequence $\left\{{\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{k}|\right)\right\}$ is bounded, since the sequences $\left\{I\left({u}_{k}\right)\right\}$ and $\left\{\parallel {u}_{k}\parallel \right\}$ are bounded. So using Fatou’s lemma and $\mathrm{ln}\left(|{u}_{k}|\right)\le {u}_{k}$, for each k, we get

$-\mathrm{\infty }<{\overline{\mathrm{lim}}}_{k\to \mathrm{\infty }}{\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{k}|\right)𝑑x\le {\int }_{\mathrm{\Omega }}{\overline{\mathrm{lim}}}_{k\to \mathrm{\infty }}\mathrm{ln}\left(|{u}_{k}|\right)dx={\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{u}_{\lambda }|\right)𝑑x.$

which implies ${u}_{\lambda }\not\equiv 0$. Thus, in both cases we have shown that ${u}_{\lambda }\in {X}_{+,q}$. So, there exist $0<{t}_{1}<{t}_{2}$ such that ${\varphi }_{{u}_{\lambda }}^{\prime }\left({t}_{1}\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left({t}_{2}\right)=0$ and ${t}_{1}{u}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{+}$. Then, the following three cases arise:

• (i)

${t}_{2}<1$,

• (ii)

${t}_{2}\ge 1$ and $\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}<0$,

• (iii)

${t}_{2}\ge 1$ and $\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}\ge 0$.

(i) Let $h\left(t\right)={\varphi }_{{u}_{\lambda }}\left(t\right)+\frac{{c}^{p}{t}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}{t}^{{p}_{s}^{*}}}{{p}_{s}^{*}}$ for $t>0$. By (4.2), we get ${h}^{\prime }\left(1\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left(1\right)+{c}^{p}-{d}^{{p}_{s}^{*}}=0$ and

${h}^{\prime }\left({t}_{2}\right)={\varphi }_{{u}_{\lambda }}^{\prime }\left({t}_{2}\right)+{t}_{2}^{p}{c}^{p}-{t}_{2}^{{p}_{s}^{*}}{d}^{{p}_{s}^{*}}=t_{2}{}^{p}\left({c}^{p}-{t}_{2}^{{p}_{s}^{*}-p}{d}^{{p}_{s}^{*}}\right)>{t}_{2}^{p}\left({c}^{p}-{d}^{{p}_{s}^{*}}\right)>0,$

which implies that h increases in $\left[{t}_{2},1\right]$. Then we get

$infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=\underset{k\to \mathrm{\infty }}{lim}I\left({u}_{k}\right)\ge {\varphi }_{u}\left(1\right)+\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}=h\left(1\right)$$>h\left({t}_{2}\right)={\varphi }_{u}\left({t}_{2}\right)+\frac{{c}^{p}{t}_{2}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}{t}_{2}^{{p}_{s}^{*}}}{{p}_{s}^{*}}$$\ge {\varphi }_{u}\left({t}_{2}\right)+\frac{{t}_{2}^{p}}{p}\left({c}^{p}-{d}^{{p}_{s}^{*}}\right)>{\varphi }_{u}\left({t}_{2}\right)$$>{\varphi }_{u}\left({t}_{1}\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right),$

(ii) In this case, we have $\left(\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}\right)<0$ and $S{d}^{p}\le {c}^{p}$. Since $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$, we have

$sup\left\{{\parallel u\parallel }^{p}:u\in {\mathcal{𝒩}}_{\lambda }^{+}\right\}\le {\left(\frac{p}{{p}_{s}^{*}}\right)}^{\frac{p}{{p}_{s}^{*}-p}}{S}^{\frac{{p}_{s}^{*}}{{p}_{s}^{*}-p}}<{c}^{p}\le sup\left\{{\parallel u\parallel }^{p}:u\in {\mathcal{𝒩}}_{\lambda }^{+}\right\},$

Consequently, only (iii) holds true, where we have

$infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)=I\left({u}_{\lambda }\right)+\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}\ge I\left({u}_{\lambda }\right)={\varphi }_{{u}_{\lambda }}\left(1\right)\ge {\varphi }_{{u}_{\lambda }}\left({t}_{1}\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right).$

Clearly, this holds when ${t}_{1}=1$ and $\left(\frac{{c}^{p}}{p}-\frac{{d}^{{p}_{s}^{*}}}{{p}_{s}^{*}}\right)=0$, which yields $c=0$ and ${u}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{+}$. Thus, ${u}_{k}\to {u}_{\lambda }$ strongly in ${X}_{0}$ as $k\to \mathrm{\infty }$ and $I\left({u}_{\lambda }\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{+}\right)$. ∎

#### Proposition 4.2.

${u}_{\lambda }$ is a positive weak solution of ().

#### Proof.

Let $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. By Lemma 3.7, since $\varphi >0$, we can find $\beta >0$ such that ${u}_{\lambda }\ge \beta$ on the support of ψ. Then ${u}_{\lambda }+ϵ\psi \ge 0$ for small ϵ. With similar reasoning as in the proof of Lemma 3.5, $I\left({u}_{\lambda }+ϵ\psi \right)\ge I\left({u}_{\lambda }\right)$ for sufficiently small $ϵ>0$. Then we have

$0\le \underset{ϵ\to 0}{lim}\frac{I\left({u}_{\lambda }+ϵ\psi \right)-I\left({u}_{\lambda }\right)}{ϵ}$$={\int }_{Q}\frac{{|{u}_{\lambda }\left(x\right)-{u}_{\lambda }\left(y\right)|}^{p-2}\left({u}_{\lambda }\left(x\right)-{u}_{\lambda }\left(y\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+ps}}𝑑x𝑑y-\lambda {\int }_{\mathrm{\Omega }}{u}_{\lambda }^{-q}\psi 𝑑x-{\int }_{\mathrm{\Omega }}{u}_{\lambda }^{\alpha }\psi 𝑑x.$

Since $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ is arbitrary, we conclude that ${u}_{\lambda }$ is a positive weak solution of (). ∎

We recall the following comparison principle from [32].

#### Lemma 4.3.

Let $u\mathrm{,}v\mathrm{\in }{X}_{\mathrm{0}}$ be such that $u\mathrm{\ge }v$ in ${\mathrm{R}}^{n}\mathrm{\setminus }\mathrm{\Omega }$ and

${\int }_{Q}\left({|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)-{|v\left(x\right)-v\left(y\right)|}^{p-2}\left(v\left(x\right)-v\left(y\right)\right)\right)\frac{\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+ps}}𝑑x𝑑y\ge 0$

for all $\mathrm{0}\mathrm{\le }\psi \mathrm{\in }{X}_{\mathrm{0}}$. Then $u\mathrm{\ge }v$ in Ω.

#### Proof.

The proof follows by taking $\psi ={\left(v-u\right)}^{+}$ and using the equality

$|b{|}^{p-2}b-|a{|}^{p-2}a=\left(p-1\right)\left(b-a\right){\int }_{0}^{1}|a+t\left(b-a\right){|}^{p-2}dt.\mathit{∎}$

As a consequence, we have the following.

#### Lemma 4.4.

We have ${\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{<}\mathrm{\infty }$.

#### Proof.

Suppose ${\mathrm{\Lambda }}_{1}=\mathrm{\infty }$. Then, from Proposition 4.2, () has a solution for all λ. Now, we choose λ large enough such that

Then $\overline{u}:={u}_{\lambda }$ is a super solution of the eigenvalue problem

(Qϵ)

Also we can choose r small enough such that $\underset{¯}{u}:=r{\varphi }_{1}$ is a subsolution of ().

Then, using the boundedness of ${u}_{\lambda }$ (see Theorem 6.4) and ${\varphi }_{1}$, we can further choose r small enough such that $\underset{¯}{u}\le \overline{u}$.

Now, we consider the following monotone iterations:

Then, by the weak comparison Lemma 4.3, we get

Therefore, the sequence $\left\{{u}_{k}\right\}$ is bounded in ${X}_{0}$, and hence has a weakly convergent subsequence, still calling it $\left\{{u}_{k}\right\}$, that converges to ${u}_{0}$ in ${X}_{0}$. Thus, ${u}_{0}$ is a weak solution of (). Since $ϵ>0$ is arbitrary, we get a contradiction to the simplicity and isolatedness of ${\lambda }_{1}$. ∎

## 5 Existence of minimizer on ${\mathcal{𝒩}}_{\lambda }^{-}$

In this section we show the existence of second solution for () in the subcritical case. We assume $\alpha +1<{p}_{s}^{*}$ throughout this section.

#### Proposition 5.1.

For all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{\right)}$, there exist ${v}_{\lambda }\mathrm{\in }{\mathcal{N}}_{\lambda }^{\mathrm{-}}$ satisfying $I\mathit{}\mathrm{\left(}{v}_{\lambda }\mathrm{\right)}\mathrm{=}\mathrm{inf}\mathit{}I\mathit{}\mathrm{\left(}{\mathcal{N}}_{\lambda }^{\mathrm{-}}\mathrm{\right)}$.

#### Proof.

Assume $0 and $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$. We will show that there exists ${v}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{-}$ with $I\left({v}_{\lambda }\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)$. Let $\left\{{v}_{k}\right\}\subset {\mathcal{𝒩}}_{\lambda }^{-}$ be a sequence such that ${lim}_{k\to \mathrm{\infty }}I\left({v}_{k}\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)$. Using Lemma 3.4, we can assume that ${v}_{k}⇀{v}_{\lambda }$ weakly as $k\to \mathrm{\infty }$ in ${X}_{0}$. We claim that ${v}_{\lambda }\in {X}_{+,q}$. When $0, if ${v}_{\lambda }=0$, then $\left\{{v}_{k}\right\}$ converges strongly to 0, which contradicts Lemma 3.4. If $q=1$, we similarly have $-\mathrm{\infty }<{\int }_{\mathrm{\Omega }}\mathrm{ln}\left(|{v}_{k}|\right)𝑑x$ as in the earlier section. So, in both cases we get ${v}_{\lambda }\in {X}_{+,q}$. Next we claim that $\left\{{v}_{k}\right\}$ converges strongly to ${v}_{\lambda }$ in ${X}_{0}$. Suppose not. Then, we may assume $\parallel {v}_{k}-{v}_{\lambda }\parallel \to d>0$ and we have the following:

• (1)

$infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)=limI\left({v}_{k}\right)\ge I\left({v}_{\lambda }\right)+\frac{{d}^{p}}{p}$,

• (2)

${\varphi }_{{v}_{k}}^{\prime }\left(1\right)=0$ and ${\varphi }_{{v}_{k}}^{\prime \prime }\left(1\right)<0$ for each k $⟹{\varphi }_{{v}_{\lambda }}^{\prime }\left(1\right)+{d}^{p}=0$ and ${\varphi }_{{v}_{\lambda }}^{\prime \prime }\left(1\right)+{d}^{p}\le 0$.

By (2), we have ${\varphi }_{{v}_{\lambda }}^{\prime }\left(1\right)<0$ and ${\varphi }_{{v}_{\lambda }}^{\prime \prime }\left(1\right)<0$. So, there exists ${t}_{2}\in \left(0,1\right)$ such that ${\varphi }_{{v}_{\lambda }}^{\prime }\left({t}_{2}\right)=0$ and ${\varphi }_{{v}_{\lambda }}^{\prime \prime }\left({t}_{2}\right)<0$. Thus, ${t}_{2}{v}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{-}$. Define $g:{ℝ}^{+}\to ℝ$ as $g\left(t\right)={\varphi }_{{v}_{\lambda }}\left(t\right)+\frac{{d}^{p}{t}^{p}}{2}$ for $t>0$. From (2), we get ${g}^{\prime }\left(1\right)=0$ and, since $0<{t}_{2}<1$, ${g}^{\prime }\left({t}_{2}\right)={d}^{p}{t}_{2}^{p-1}>0$. So, g is increasing on $\left[{t}_{2},1\right]$. Now, we obtain

$infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)\ge I\left({v}_{\lambda }\right)+\frac{{d}^{p}}{p}={\varphi }_{{v}_{\lambda }}\left(1\right)+\frac{{d}^{p}}{p}=g\left(1\right)\ge g\left({t}_{2}\right)>{\varphi }_{{v}_{\lambda }}\left({t}_{2}\right)=I\left({t}_{2}{v}_{\lambda }\right)\ge infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right),$

which gives a contradiction. Hence, $d=0$, and thus $\left\{{v}_{k}\right\}$ converges strongly to ${v}_{\lambda }$ in ${X}_{0}$. Since $\lambda \in \left(0,{\mathrm{\Lambda }}_{1}\right)$, we have ${\varphi }_{{v}_{\lambda }}^{\prime \prime }\left(1\right)<0$. Therefore, we obtain ${v}_{\lambda }\in {\mathcal{𝒩}}_{\lambda }^{-}$ and $I\left({v}_{\lambda }\right)=infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)$. This completes the proof of this proposition for the subcritical case. ∎

#### Proposition 5.2.

For $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\Lambda }}_{\mathrm{1}}\mathrm{\right)}$, ${v}_{\lambda }$ is a positive weak solution of ().

#### Proof.

Let $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. Using Lemma 3.7, since $\varphi >0$ in Ω, we can find $\beta >0$ such that ${v}_{\lambda }\ge \beta$ on the support of ψ. Also, ${t}_{ϵ}\to 1$ as $ϵ\to 0+$, where ${t}_{ϵ}$ is the unique positive real number corresponding to $\left({v}_{\lambda }+ϵ\psi \right)$ such that ${t}_{ϵ}\left({v}_{\lambda }+ϵ\psi \right)\in {\mathcal{𝒩}}_{\lambda }^{-}$. Then, by Lemma 3.5, we have

$0\le \underset{ϵ\to 0}{lim}\frac{I\left({t}_{ϵ}\left({v}_{\lambda }+ϵ\psi \right)\right)-I\left({v}_{\lambda }\right)}{ϵ}\le \underset{ϵ\to 0}{lim}\frac{I\left({t}_{ϵ}\left({v}_{\lambda }+ϵ\psi \right)\right)-I\left({t}_{ϵ}{v}_{\lambda }\right)}{ϵ}$$={\int }_{Q}\frac{{|{v}_{\lambda }\left(x\right)-{v}_{\lambda }\left(y\right)|}^{p-2}\left({v}_{\lambda }\left(x\right)-{v}_{\lambda }\left(y\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {v}_{\lambda }^{-q}+{v}_{\lambda }^{\alpha }\right)\psi 𝑑x.$

Since $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ is arbitrary, we conclude that ${v}_{\lambda }$ is a positive weak solution of (). ∎

#### Proof of Theorem 2.4.

The proof follows from Propositions 4.2 and 5.2. ∎

#### Remark 5.3.

To prove the existence of second positive solution in the critical case, one requires to know the classification of exact solutions of the problem

These are the minimizers of S, the best constant of the embedding ${X}_{0}$ into ${L}^{{p}_{s}^{*}}$. In [6, 38], several estimates on these minimizers were obtained, and it was conjectured that the solutions are dilations and translations of the radial function

$U\left(x\right)=\frac{1}{{\left(1+{|x|}^{{p}^{\prime }}\right)}^{\frac{N-sp}{p}}},x\in {ℝ}^{n},$

where ${p}^{\prime }=\frac{p}{p-1}$. In the case $p=2$, these classifications were proved in [41], where, in addition, it was proved that all solutions are classified by dilations and translations of $U\left(x\right)$. Using these classifications, in [39], it was shown that

$sup\left\{I\left({u}_{\lambda }+t{U}_{ϵ}\right):t\ge 0\right\}

where ${U}_{ϵ}={ϵ}^{-\frac{n-2s}{2}}U\left(\frac{x}{ϵ}\right)$, $x\in {ℝ}^{n}$, $ϵ>0$ and ${u}_{\lambda }$ is the minimizer on ${\mathcal{𝒩}}_{\lambda }^{+}$. Then, by carefully analysing the related fiber maps, it was shown that ${u}_{\lambda }+t{U}_{ϵ}\in {\mathcal{𝒩}}_{\lambda }^{-}$ for large t. From this it follows

$infI\left({\mathcal{𝒩}}_{\lambda }^{-}\right)

Then the existence of a minimizer of ${N}_{\lambda }^{-}$ was shown using the analysis of fibering maps in Lemma 3.2.

## 6 Regularity of weak solutions

In this section, we shall prove some regularity properties of positive weak solutions of (). We begin with the following lemma.

#### Lemma 6.1.

Suppose u is a weak solution of (). Then, for each $w\mathrm{\in }{X}_{\mathrm{0}}$, ${u}^{\mathrm{-}q}\mathit{}w\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right)w𝑑x=0.$(6.1)

#### Proof.

Let u be a weak solution of () and $w\in {X}_{+}$. By Lemma 2.2, we get a sequence $\left\{{w}_{k}\right\}\in {X}_{0}$ such that $\left\{{w}_{k}\right\}\to w$ strongly in ${X}_{0}$. Each ${w}_{k}$ has compact support in Ω and $0\le {w}_{1}\le {w}_{2}\le \mathrm{\cdots }$. Therefore, u is a positive weak solution of () and, for each k, we get

$\lambda {\int }_{\mathrm{\Omega }}{u}^{-q}{w}_{k}𝑑x={\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({w}_{k}\left(x\right)-{w}_{k}\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}{u}^{\alpha }{w}_{k}𝑑x.$

Using the monotone convergence theorem, we get ${u}^{-q}w\in {L}^{1}\left(\mathrm{\Omega }\right)$ and

$\lambda {\int }_{\mathrm{\Omega }}{u}^{-q}w𝑑x={\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}{u}^{\alpha }w𝑑x.$

If $w\in {X}_{0}$, then $w={w}^{+}-{w}^{-}$ and ${w}^{+},{w}^{-}\in {X}_{+}$. Since we proved the lemma for each $w\in {X}_{+}$, we obtain the conclusion. ∎

Before proving our next result, let us recall some estimates or inequalities from [7].

#### Lemma 6.2.

Let $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty }$ and $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be a ${\text{𝐶}}^{\mathrm{1}}$ convex function. If $\tau \mathrm{\ge }\mathrm{0}$, $t\mathrm{,}a\mathrm{,}b\mathrm{\in }\mathrm{R}$ and $A\mathrm{,}B\mathrm{>}\mathrm{0}$, then

${|f\left(a\right)-f\left(b\right)|}^{p-2}\left(f\left(a\right)-f\left(b\right)\right)\left(A-B\right)\le {|a-b|}^{p-2}\left(a-b\right)\left(A{|{f}^{\prime }\left(a\right)|}^{p-2}{f}^{\prime }\left(a\right)-B{|{f}^{\prime }\left(b\right)|}^{p-2}{f}^{\prime }\left(b\right)\right).$

#### Lemma 6.3.

Let $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty }$ and $g\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be an increasing function. Then we have

${|G\left(a\right)-G\left(b\right)|}^{p}\le {|a-b|}^{p-2}\left(a-b\right)\left(g\left(a\right)-g\left(b\right)\right),$

where $G\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{\mathrm{0}}^{t}{g}^{\mathrm{\prime }}\mathit{}{\mathrm{\left(}\tau \mathrm{\right)}}^{\frac{\mathrm{1}}{p}}\mathit{}𝑑\tau$ for $t\mathrm{\in }\mathrm{R}$.

#### Theorem 6.4.

Let u be a positive solution of (). Then $u\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$.

#### Proof.

The proof here is adopted from Brasco and Parini [7]. Let $ϵ>0$ be very small and define

${f}_{ϵ}\left(t\right)={\left({ϵ}^{2}+{t}^{2}\right)}^{\frac{1}{2}},$

which is smooth, convex and Lipschitz. Let $0<\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and take $\phi =\psi {|{f}_{ϵ}^{\prime }\left(u\right)|}^{p-2}{f}_{ϵ}^{\prime }\left(u\right)$ as the test function in (6.1). By taking the choices

$a=u\left(x\right),b=u\left(y\right),A=\psi \left(x\right),B=\psi \left(y\right)$

in Lemma 6.2, we get

${\int }_{Q}\frac{{|{f}_{ϵ}\left(u\left(x\right)\right)-{f}_{ϵ}\left(u\left(y\right)\right)|}^{p-2}\left({f}_{ϵ}\left(u\left(x\right)\right)-{f}_{ϵ}\left(u\left(y\right)\right)\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y\le {\int }_{\mathrm{\Omega }}\left(|\lambda {u}^{-q}+{u}^{\alpha }|\right){|{f}_{ϵ}^{\prime }\left(u\right)|}^{p-1}\psi 𝑑x.$

As $t\to 0$, ${f}_{ϵ}\left(t\right)\to |t|$ and we have $|{f}_{ϵ}^{\prime }\left(t\right)|\le 1$. So using Fatou’s lemma, we let $ϵ\to 0$ in the above inequality to get

${\int }_{Q}\frac{{||u\left(x\right)|-|u\left(y\right)||}^{p-2}\left(|u\left(x\right)|-|u\left(y\right)|\right)\left(\psi \left(x\right)-\psi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y\le {\int }_{\mathrm{\Omega }}\left(|\lambda {u}^{-q}+{u}^{\alpha }|\right)\psi 𝑑x$(6.2)

for every $0<\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. The above inequality still holds for $0\le \psi \in {X}_{0}$ ( similar proof as of Lemma 6.1). Now, define ${u}_{K}=\mathrm{min}\left\{{\left(u-1\right)}^{+},K\right\}\in {X}_{0}$ for $K>0$. For $\beta >0$ and $\rho >0$, we take $\psi ={\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }$ as test function in (6.2) and get

${\int }_{Q}\frac{|u\left(x\right)|-{|u\left(y\right)|}^{p-2}\left(|u\left(x\right)|-|u\left(y\right)|\right)\left({\left({u}_{K}\left(x\right)+\rho \right)}^{\beta }-{\left({u}_{K}\left(y\right)+\rho \right)}^{\beta }\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y\le {\int }_{\mathrm{\Omega }}\left(|\lambda {u}^{-q}+{u}^{\alpha }|\right)\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)𝑑x.$

Then, using Lemma 6.3 with the function $g\left(u\right)={\left({u}_{K}+\rho \right)}^{\beta }$, we get

${\int }_{Q}\frac{|{\left({u}_{K}\left(x\right)+\rho \right)}^{\frac{\beta +p-1}{p}}-{\left({u}_{K}\left(y\right)+\rho \right)}^{\frac{\beta +p-1}{p}}|{}^{n+sp}}{x-y}𝑑x𝑑y$$\le \frac{{\left(\beta +p-1\right)}^{p}}{\beta {p}^{p}}{\int }_{Q}\frac{{||u\left(x\right)|-|u\left(y\right)||}^{p-2}\left(|u\left(x\right)|-|u\left(y\right)|\right)\left({\left({u}_{K}\left(x\right)+\rho \right)}^{\beta }-{\left({u}_{K}\left(y\right)+\rho \right)}^{\beta }\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$\le \frac{{\left(\beta +p-1\right)}^{p}}{\beta {p}^{p}}{\int }_{\mathrm{\Omega }}\lambda |{u}^{-q}|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx+{\int }_{\mathrm{\Omega }}|{u}^{\alpha }|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx.$

Now, from the support of ${u}_{K}$, we have

${\int }_{\mathrm{\Omega }}\lambda |{u}^{-q}|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx+{\int }_{\mathrm{\Omega }}|{u}^{\alpha }|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx$$={\int }_{\left\{u\ge 1\right\}}\lambda |{u}^{-q}|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx+{\int }_{\left\{u\ge 1\right\}}|{u}^{\alpha }|\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx$$\le {C}_{1}{\int }_{\left\{u\ge 1\right\}}\left(1+{|u|}^{\alpha }\right)\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)𝑑x$$\le 2{C}_{1}{\int }_{\left\{u\ge 1\right\}}|u{|}^{\alpha }\left({\left({u}_{K}+\rho \right)}^{\beta }-{\rho }^{\beta }\right)dx$$\le 2{C}_{1}{|u|}_{{p}_{s}^{*}}^{\alpha }{|{\left({u}_{K}+\rho \right)}^{\beta }|}_{r},$

where ${C}_{1}=\mathrm{max}\left\{\lambda ,1\right\}$ and $r=\frac{{p}_{s}^{*}}{{p}_{s}^{*}-\alpha }$. Using the Sobolev inequality, given in [33, Theorem 1], we get

${\int }_{Q}\frac{|{\left({u}_{K}\left(x\right)+\rho \right)}^{\frac{\beta +p-1}{p}}-{\left({u}_{K}\left(y\right)+\rho \right)}^{\frac{\beta +p-1}{p}}|}{x-{y}^{n+sp}}𝑑x𝑑y\ge \frac{1}{{T}_{p,s}}{|{\left({u}_{K}+\rho \right)}^{\frac{\beta +p-1}{p}}-{\rho }^{\frac{\beta +p-1}{p}}|}_{{p}_{s}^{*}}^{p}$$\ge \frac{1}{{T}_{p,s}}\left({\left(\frac{\rho }{2}\right)}^{p-1}{|{\left({u}_{K}+\rho \right)}^{{\beta }_{p}}|}_{{p}_{s}^{*}}^{p}-{\rho }^{\beta +p-1}{|\mathrm{\Omega }|}^{\frac{p}{{p}_{s}^{*}}}\right),$

where ${T}_{p,s}$ is a nonnegative constant and the last inequality follows from the triangle inequality and ${\left({u}_{K}+\rho \right)}^{\beta +p-1}\ge {\rho }^{p-1}{\left({u}_{K}+\rho \right)}^{\beta }$. Using all these estimates, we now have

${|{\left({u}_{K}+\rho \right)}^{\frac{\beta }{p}}|}_{{p}_{s}^{*}}^{p}\le C\left({T}_{p,s}{\left(\frac{2}{\rho }\right)}^{p-1}\left(\frac{{\left(\beta +p-1\right)}^{p}}{\beta {p}^{p}}\right){|u|}_{{p}_{s}^{*}}^{\alpha }{|{\left({u}_{K}+\rho \right)}^{\beta }|}_{r}+{\rho }^{\beta }{|\mathrm{\Omega }|}^{\frac{p}{{p}_{s}^{*}}}\right),$

where $C=C\left(p\right)>0$ is a constant. By the convexity of the map $t↦{t}^{p}$, we can show that

$\frac{1}{\beta }{\left(\frac{\beta +p-1}{p}\right)}^{p}\ge 1.$

Using this, we can also check that

${\rho }^{\beta }{|\mathrm{\Omega }|}^{\frac{p}{{p}_{s}^{*}}}\le \frac{1}{\beta }{\left(\frac{\beta +p-1}{p}\right)}^{p}{|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}{|{\left({u}_{K}+\rho \right)}^{\beta }|}_{r}.$

Hence, we have

${|{\left({u}_{K}+\rho \right)}^{\frac{\beta }{p}}|}_{{p}_{s}^{*}}^{p}\le C\frac{1}{\beta }{\left(\frac{\beta +p-1}{p}\right)}^{p}{|{\left({u}_{K}+\rho \right)}^{\beta }|}_{r}\left(\frac{{T}_{p,s}{|u|}_{{p}_{s}^{*}}^{\alpha }}{{\rho }^{p-1}}+{|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)$

for $C=C\left(p\right)>0$ constant. We now choose

$\rho ={\left({T}_{p,s}{|u|}_{{p}_{s}^{*}}^{\alpha }\right)}^{\frac{1}{p-1}}{|\mathrm{\Omega }|}^{\frac{-1}{p-1}\left(1-\frac{1}{r}-\frac{sp}{n}\right)},$

and let $\beta \ge 1$ be such that

$\frac{1}{\beta }{\left(\frac{\beta +p-1}{p}\right)}^{p}\le {\beta }^{p-1}.$

In addition, if we let $\tau =\beta r$ and $\nu =\frac{{p}_{s}^{*}}{pr}>1$, then the above inequality reduces to

${|\left({u}_{K}+\rho \right)|}_{\nu \tau }\le {\left(C{|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)}^{\frac{r}{\tau }}{\left(\frac{\tau }{r}\right)}^{\frac{\left(p-1\right)r}{\tau }}{|\left({u}_{K}+\rho \right)|}_{\tau }.$(6.3)

At this stage, if we take $K\to \mathrm{\infty }$, then we can say that ${\left(u-1\right)}^{+}\in {L}^{m}\left(\mathrm{\Omega }\right)$ for all m. This will imply that $u\in {L}^{m}\left(\mathrm{\Omega }\right)$ for all m. Now, we iterate (6.3) using ${\tau }_{0}=r$, and let ${\tau }_{m+1}=\nu {\tau }_{m}={\nu }^{m+1}r,$ which gives

${|\left({u}_{K}+\rho \right)|}_{{\tau }_{m+1}}\le {\left(C{|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)}^{{\sum }_{i=0}^{m}\frac{r}{{\tau }_{i}}}{\left(\prod _{i=0}^{m}{\left(\frac{{\tau }_{i}}{r}\right)}^{\frac{r}{{\tau }_{i}}}\right)}^{p-1}{|\left({u}_{K}+\rho \right)|}_{r}.$(6.4)

Since $\nu >1$,

$\sum _{i=0}^{\mathrm{\infty }}\frac{r}{{\tau }_{i}}=\sum _{i=0}^{m}\frac{1}{{\nu }^{i}}=\frac{\nu }{\nu -1}\mathit{ }\text{and}\mathit{ }\prod _{i=0}^{\mathrm{\infty }}{\left({\left(\frac{{\tau }_{i}}{r}\right)}^{\frac{r}{{\tau }_{i}}}\right)}^{p-1}={\nu }^{\frac{\nu }{{\left(\nu -1\right)}^{2}}}.$

Taking limit as $m\to 0$ in (6.4), we finally get

${|{u}_{K}|}_{\mathrm{\infty }}\le {\left(C{\nu }^{\frac{\nu }{{\left(\nu -1\right)}^{2}}}\right)}^{p-1}{\left({|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)}^{\frac{\nu }{\nu -1}}{|\left({u}_{K}+\rho \right)|}_{r}.$

Since ${u}_{K}\le {\left(u-1\right)}^{+}$, using the triangle inequality in the above inequality, we get

${|{u}_{K}|}_{\mathrm{\infty }}\le C{\left({\nu }^{\frac{\nu }{{\left(\nu -1\right)}^{2}}}\right)}^{p-1}{\left({|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)}^{\frac{\nu }{\nu -1}}\left({|{\left(u-1\right)}^{+}|}_{r}+\rho {|\mathrm{\Omega }|}^{\frac{1}{r}}\right)$

for some constant $C=C\left(p\right)>0$. If we now let $K\to \mathrm{\infty }$, we get

${|{\left(u-1\right)}^{+}|}_{\mathrm{\infty }}\le C{\left({\nu }^{\frac{\nu }{{\left(\nu -1\right)}^{2}}}\right)}^{p-1}{\left({|\mathrm{\Omega }|}^{1-\frac{1}{r}-\frac{sp}{n}}\right)}^{\frac{\nu }{\nu -1}}\left({|{\left(u-1\right)}^{+}|}_{r}+\rho {|\mathrm{\Omega }|}^{\frac{1}{r}}\right).$

Hence, in particular, we have that $u\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. ∎

#### Theorem 6.5.

Let u be a positive solution of (). Then there exists $\gamma \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}s\mathrm{\right]}$ such that $u\mathrm{\in }{C}_{\mathrm{loc}}^{\gamma }\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$.

#### Proof.

Let ${\mathrm{\Omega }}^{\prime }\subset \subset \mathrm{\Omega }$. Then, using Lemma 3.7 and the above regularity result, for any $\psi \in {C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, we get

$\lambda {\int }_{{\mathrm{\Omega }}^{\prime }}{u}^{-q}\psi 𝑑x+{\int }_{{\mathrm{\Omega }}^{\prime }}{u}^{\alpha }\psi 𝑑x\le \lambda {\int }_{{\mathrm{\Omega }}^{\prime }}{\varphi }_{1}^{-q}\psi 𝑑x+{\parallel u\parallel }_{\mathrm{\infty }}^{\alpha }{\int }_{{\mathrm{\Omega }}^{\prime }}\psi 𝑑x\le C{\int }_{{\mathrm{\Omega }}^{\prime }}\psi 𝑑x$

for some constant $C>0$, since we can find $k>0$ such that ${\varphi }_{1}>k$ on ${\mathrm{\Omega }}^{\prime }$. Thus, we have $|{\left(-{\mathrm{\Delta }}_{p}\right)}^{s}u|\le C$ weakly in ${\mathrm{\Omega }}^{\prime }$. So, using [31, Theorem 4.4] and applying a covering argument on the above inequality as in [31, Corollary 5.5], we can prove that there exists $\gamma \in \left(0,s\right]$ such that $u\in {C}^{\gamma }\left({\mathrm{\Omega }}^{\prime }\right)$ for all ${\mathrm{\Omega }}^{\prime }⋐\mathrm{\Omega }$. Therefore, we get $u\in {C}_{\mathrm{loc}}^{\gamma }\left(\mathrm{\Omega }\right)$. ∎

## 7 Global existence of solution

In this section, we show the existence of solution for maximal range of λ. Let us define

#### Lemma 7.1.

We have $\mathrm{\Lambda }\mathrm{<}\mathrm{+}\mathrm{\infty }$.

#### Proof.

The proof follows similarly to the proof of Lemma 4.4. ∎

In the following lemmas, we will show the existence of a solution of () when $\lambda \in \left(0,\mathrm{\Lambda }\right)$.

#### Lemma 7.2.

If $\underset{\mathrm{¯}}{u}\mathrm{\in }{X}_{\mathrm{0}}$ is a weak sub-solution and $\overline{u}\mathrm{\in }{X}_{\mathrm{0}}$ is a weak super-solution of () such that $\underset{\mathrm{¯}}{u}\mathrm{\le }\overline{u}$ a.e. in Ω, then there exists a weak solution $u\mathrm{\in }{X}_{\mathrm{0}}$ satisfying $\underset{\mathrm{¯}}{u}\mathrm{\le }u\mathrm{\le }\overline{u}$.

#### Proof.

We follow [28]. We know that the functional I is non- differentiable in ${X}_{0}$. Let $M:=\left\{u\in {X}_{0}:\underset{¯}{u}\le u\le \overline{u}\right\}$. Then M is closed and convex, and I is weakly lower semicontinuous on M. We can see that if $\left\{{u}_{k}\right\}\subset M$ and ${u}_{k}⇀u$ weakly in ${X}_{0}$ as $k\to \mathrm{\infty }$, then we may assume ${u}_{k}\to u$ pointwise a.e. in Ω (along a subsequence). Since $u\in M$, ${\int }_{\mathrm{\Omega }}|\overline{u}{|}^{\alpha +1}dx<+\mathrm{\infty }$ and ${\int }_{\mathrm{\Omega }}{G}_{q}\left(\overline{u}\right)𝑑x<+\mathrm{\infty }$, so, by Lebesgue’s dominated convergence theorem,

So, ${\underset{¯}{\mathrm{lim}}}_{k\to \mathrm{\infty }}I\left({u}_{k}\right)\ge I\left(u\right)$. Thus, there exists $u\in M$ such that $I\left(u\right)=infI\left(M\right)$. We claim that u is a weak solution of (). For $ϵ>0$ and $\phi \in {X}_{0}$, define ${v}_{ϵ}=u+ϵ\phi -{\phi }^{ϵ}+{\phi }_{ϵ}\in M$, where ${\phi }^{ϵ}={\left(u+ϵ\phi -\overline{u}\right)}^{+}\ge 0$ and ${\phi }_{ϵ}={\left(u+ϵ\phi -\underset{¯}{u}\right)}^{-}\ge 0$. For $t\in \left(0,1\right)$, $u+t\left({v}_{ϵ}-u\right)\in M$ and we have

$0\le \frac{I\left(u+t\left({v}_{ϵ}-u\right)\right)-I\left(u\right)}{t}$$=\underset{t\to 0}{lim}\left(\frac{1}{pt}\left(\parallel u+t\left({v}_{ϵ}-u\right){\parallel }^{p}-\parallel u{\parallel }^{p}\right)+\lambda {\int }_{\mathrm{\Omega }}\frac{\left({G}_{q}\left(u+t\left({v}_{ϵ}-u\right)\right)-{G}_{q}\left(u\right)\right)}{t}dx$$-\frac{1}{\alpha +1}{\int }_{\mathrm{\Omega }}\frac{{|u+t\left({v}_{ϵ}-u\right)|}^{\alpha +1}-{|u|}^{\alpha +1}}{t}dx\right)$$={\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\left({v}_{ϵ}-u\right)\left(x\right)-\left({v}_{ϵ}-u\right)\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$-\lambda {\int }_{\mathrm{\Omega }}{u}^{-q}\left({v}_{ϵ}-u\right)𝑑x-{\int }_{\mathrm{\Omega }}{u}^{\alpha }\left({v}_{ϵ}-u\right)𝑑x,$

which gives

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right)\phi 𝑑x\ge \frac{1}{ϵ}\left({H}^{ϵ}-{H}_{ϵ}\right),$(7.1)

where

${H}^{ϵ}={\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({\phi }^{ϵ}\left(x\right)-{\phi }^{ϵ}\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){\phi }^{ϵ}𝑑x,$${H}_{ϵ}={\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({\phi }_{ϵ}\left(x\right)-{\phi }_{ϵ}\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){\phi }_{ϵ}𝑑x.$

Now, we consider

$\frac{1}{ϵ}{H}^{ϵ}=\frac{1}{ϵ}\left({\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({\phi }^{ϵ}\left(x\right)-{\phi }^{ϵ}\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){\phi }^{ϵ}𝑑x\right).$

Let ${\mathrm{\Omega }}_{1}=\left\{u+ϵ\phi \ge \overline{u}>u\right\}$ and ${\mathrm{\Omega }}_{2}=\left\{u+ϵ\phi <\underset{¯}{u}\right\}$. Then, using the technique of Lemma 3.7, we get

$\frac{1}{ϵ}{\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({\phi }^{ϵ}\left(x\right)-{\phi }^{ϵ}\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$=\frac{1}{ϵ}\left({\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}+{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\right)\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left({\phi }^{ϵ}\left(x\right)-{\phi }^{ϵ}\left(y\right)\right)}{{|x-y|}^{n+sp}}dxdy$$=\frac{1}{ϵ}{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\left(u-\overline{u}\right)\left(x\right)-\left(u-\overline{u}\right)\left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$+\frac{1}{ϵ}{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{n+sp}}\left(u-\overline{u}\right)\left(x\right)𝑑x𝑑y$$+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{n+sp}}\phi \left(x\right)𝑑x𝑑y$$-\frac{1}{ϵ}{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{n+sp}}\left(u-\overline{u}\right)\left(y\right)𝑑x𝑑y$$-{\int }_{{\mathrm{\Omega }}_{2}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{n+sp}}\phi \left(y\right)𝑑x𝑑y$$\ge \frac{3}{ϵ{2}^{p-2}}{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|\left(u-\overline{u}\right)\left(x\right)-\left(u-\overline{u}\right)\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}𝑑x𝑑y$$+{\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y$$\ge {\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y,$

where we used the inequality ${|a-b|}^{p}\le {2}^{p-2}\left({|a|}^{p-2}a-{|b|}^{p-2}b\right)\left(a-b\right)$ for $p\ge 2$ and $a,b\in ℝ$. Thus,

$\frac{1}{ϵ}{H}^{ϵ}\ge {\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{{\mathrm{\Omega }}_{1}}\left(\lambda {u}^{-q}+{u}^{\alpha }\right){\phi }^{ϵ}𝑑x$$\ge {\int }_{{\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{1}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}dxdy-{\int }_{{\mathrm{\Omega }}_{1}}|\lambda {\overline{u}}^{-q}-{u}^{-q}||\phi |dx$

since $|{\mathrm{\Omega }}_{1}|\to 0$ as $ϵ\to 0$. Similarly, as $ϵ\to 0$, we can show that

$\frac{1}{ϵ}{H}_{ϵ}\le o\left(1\right).$

Therefore, taking $ϵ\to 0$ in (7.1), we get

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right)\phi 𝑑x\ge o\left(1\right).$

Since $\phi \in {X}_{0}$ is arbitrary, for all $\phi \in {X}_{0}$, we get

${\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(\phi \left(x\right)-\phi \left(y\right)\right)}{{|x-y|}^{n+sp}}𝑑x𝑑y-{\int }_{\mathrm{\Omega }}\left(\lambda {u}^{-q}+{u}^{\alpha }\right)\phi 𝑑x=0.\mathit{∎}$

#### Proposition 7.3.

For $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{\Lambda }\mathrm{\right)}$, () has a weak solution ${u}_{\lambda }\mathrm{\in }{X}_{\mathrm{0}}$.

#### Proof.

We fix $\lambda \in \left(0,\mathrm{\Lambda }\right)$. By the definition of Λ, there exists ${\lambda }_{0}\in \left(\lambda ,\mathrm{\Lambda }\right)$ such that (P${}_{{\lambda }_{0}}$) has a solution ${u}_{{\lambda }_{0}}$, say. Then $\overline{u}={u}_{{\lambda }_{0}}$ becomes a super-solution of (). Now, consider the function ${\varphi }_{1}$ as the eigenfunction of ${\left(-{\mathrm{\Delta }}_{p}\right)}^{s}$ corresponding to the smallest eigenvalue ${\lambda }_{1}$. Then ${\varphi }_{1}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and

Let us choose $t>0$ such that $t{\varphi }_{1}\le \overline{u}$ and ${t}^{p+q-1}{\varphi }_{1}^{p+q-1}\le \frac{\lambda }{{\lambda }_{1}}$. If we define $\underset{¯}{u}=t{\varphi }_{1}$, then

${\left(-{\mathrm{\Delta }}_{p}\right)}^{s}\underset{¯}{u}={\lambda }_{1}{t}^{p-1}{\varphi }_{1}^{p-1}\le \lambda {t}^{-q}{\varphi }_{1}^{-q}\le \lambda {t}^{-q}{\varphi }_{1}^{-q}+{t}^{\alpha }{\varphi }_{1}^{\alpha }=\lambda {\underset{¯}{u}}^{-q}+{\underset{¯}{u}}^{\alpha },$

that is, $\underset{¯}{u}$ is a sub-solution of () and $\underset{¯}{u}\le \overline{u}$. Applying Lemma 7.2, we conclude that () has a solution for all $\lambda \in \left(0,\mathrm{\Lambda }\right)$. This completes the proof. ∎

#### Proof of Theorem 2.6.

The proof follows from Proposition 7.3 and Lemma 7.1. ∎

#### Remark 7.4.

We remark that using the method in Lemma 7.2, we can show the existence of solution for the following pure singular problem:

(7.2)

where $0. We define u to be a positive weak solution of (7.2) if $u>0$ in Ω, $u\in {X}_{0}$ and

Also, we say that $u\in {X}_{0}$ is a positive weak sub-solution of (7.2) if $u>0$ and

We define the functional ${J}_{\lambda }:{X}_{0}\to \left(-\mathrm{\infty },\mathrm{\infty }\right]$ by

${J}_{\lambda }\left(u\right)=\frac{1}{p}{\int }_{Q}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{n+sp}}𝑑x𝑑y-\lambda {\int }_{\mathrm{\Omega }}{G}_{q}\left(u\right)𝑑x,$

where ${G}_{q}$ is as defined in Section 2. One can easily see that ${J}_{\lambda }$ is coercive, bounded below and weakly lower semicontinuous in ${X}_{0}$. Thus, there exist ${u}_{0}\in {X}_{0}$ such that $infI\left({X}_{0}\right)=I\left({u}_{0}\right)$. We claim that ${u}_{0}$ is a positive weak solution of (7.2). We choose $t>0$ such that $t{\varphi }_{1}\le {u}_{0}$ in Ω and $t{\varphi }_{1}$ is a sub-solution of (7.2) (${\varphi }_{1}$ is defined in Proposition 7.3). Let us define $M:=\left\{u\in {X}_{0}:\underset{¯}{u}\le u\right\}$, where $\underset{¯}{u}$ is a weak sub-solution of (7.2). Then ${u}_{0}\in M$, and following the proof of Lemma 7.2 with ${v}_{ϵ}={u}_{0}+ϵ\phi +{\phi }_{ϵ}$, where $ϵ>0$, ${\phi }_{ϵ}={\left({u}_{0}+ϵ\phi -\underset{¯}{u}\right)}^{-}$ and $\phi \in {X}_{0}$, we can show that ${u}_{0}$ is a positive weak solution of (7.2).

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Revised: 2016-07-26

Accepted: 2016-09-18

Published Online: 2016-12-02

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 52–72, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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