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Volume 7, Issue 4

# On fractional p-Laplacian problems with local conditions

Anran Li
/ Chongqing Wei
Published Online: 2016-11-23 | DOI: https://doi.org/10.1515/anona-2016-0105

## Abstract

In this paper, we deal with fractional p-Laplacian equations of the form

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}u& =\lambda f\left(x,u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill u\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega },\end{array}$

where $\lambda \in \left(0,+\mathrm{\infty }\right)$, $0 and $\mathrm{\Omega }\subset {ℝ}^{N}$, $N⩾2$, is a bounded domain with smooth boundary. With assumptions on $f\left(x,t\right)$ just in $\mathrm{\Omega }×\left(-\delta ,\delta \right)$, where $\delta >0$ is small, existence and multiplicity of nontrivial solutions are obtained via variational methods.

Keywords: local conditions; variational methods.

MSC 2010: 35R11; 35A15; 35B38

## 1 Introduction and main results

In this paper we are concerned with the existence and multiplicity of nontrivial solutions of the following fractional p-Laplacian equation:

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}u& =\lambda f\left(x,u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill u\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega },\end{array}$(Pλ)

where $\lambda \in \left(0,+\mathrm{\infty }\right)$, $0, $\mathrm{\Omega }\subset {ℝ}^{N}$ is a bounded domain with smooth boundary, $N⩾2$, and $f\left(x,t\right)$ is a Carathéodory function defined on $\mathrm{\Omega }×\left(-\delta ,\delta \right)$, with $\delta >0$ being small.

When $p=2$, much attention has been paid to the semi-linear problem

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}^{s}u& =g\left(x,u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill u\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega },\end{array}$

from the point of view of existence, non-existence and regularity, where $g:\mathrm{\Omega }×ℝ↦ℝ$ is a Carathéodory function satisfying suitable growth conditions. Several existence results via variational methods are proved in a series of papers [18, 20, 19, 21]. The issues of regularity and non-existence of solutions are studied in [4, 15, 16, 14, 3]. The corresponding equations in ${ℝ}^{N}$ have also been widely studied, see, for example [17, 5, 9, 1, 10] and references therein.

Very recently, a new nonlocal and nonlinear operator was considered, namely, for $p\in \left(1,+\mathrm{\infty }\right)$, $s\in \left(0,1\right)$ and u smooth enough,

${\left(-\mathrm{\Delta }\right)}_{p}^{s}u\left(x\right)=2\underset{\epsilon \to {0}^{+}}{lim}{\int }_{{ℝ}^{N}\setminus {B}_{\epsilon }\left(x\right)}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{N+sp}}𝑑y,x\in {ℝ}^{N},$

which is consistent, up to some normalization constant depending upon N and s, with the linear fractional Laplacian ${\left(-\mathrm{\Delta }\right)}^{s}$ in the case $p=2$. This operator, known as the fractional p-Laplacian, leads naturally to the quasilinear problem (P${}_{1}$). One typical feature of this operator is the nonlocality, in the sense that the value of ${\left(-\mathrm{\Delta }\right)}_{p}^{s}u$ at any point $x\in \mathrm{\Omega }$ depends not only on the values of u in Ω, but actually on the whole ${ℝ}^{N}$, since $u\left(x\right)$ represents the expected value of a random variable ties to a process randomly jumping arbitrarily far from the point x. While in the classical case, by the continuity properties of the Brownian motion, at the exit time from Ω one necessarily is on $\partial \mathrm{\Omega }$, due to the jumping nature of the process, at the exit time one could end up anywhere outside Ω. In this sense, the natural nonhomogeneous Dirichlet boundary condition consists in assigning the values of u in ${ℝ}^{N}\setminus \mathrm{\Omega }$ rather than merely on $\partial \mathrm{\Omega }$. Then, it is reasonable to search for solution in the space of functions $u\in {W}^{s,p}\left({ℝ}^{N}\right)$ vanishing outside Ω. It should be pointed out that in a bounded domain, this is not the only possible way to provide a formulation for the problem. In the works of [11, 13], the eigenvalue problem associated with ${\left(-\mathrm{\Delta }\right)}_{p}^{s}$ is studied, and particularly some properties of the first eigenvalue and of the higher order (variational) eigenvalues are obtained. From the point of view of regularity theory, some results can be found in [13]. This work is most focused on the case where p is large and the solutions inherit some regularity directly from the functional embedding theorems. In [7, 2], relevant results about the local boundedness and Hölder continuity for the solutions to the problem of finding $\left(s,p\right)$-harmonic functions u were obtained. Very recently, in [12], Iannizzotto et al. established a priori ${L}^{\mathrm{\infty }}$-bounds for the solutions of problem (P${}_{1}$) under suitable growth conditions on the nonlinearity.

In all the works mentioned above, the nonlinearity is assumed on the whole $\mathrm{\Omega }×ℝ$. Motivated by [12, 22, 6], we can consider problem () with local conditions on the nonlinearity f. Firstly, we assume that f satisfies a p-sublinear condition at the origin, without any growth condition at infinity. In particular, we assume the following:

• (f0)

$f\left(x,t\right)$ is a Carathéodory function defined on $\mathrm{\Omega }×\left(-\delta ,\delta \right)$.

• (f1)

There exists positive constant ${p}_{1}\in \left(\frac{{p}^{2}}{{p}_{s}^{*}},p\right)$ such that

where $F\left(x,t\right)={\int }_{0}^{t}f\left(x,\xi \right)𝑑\xi$.

• (f2)

There exists positive constant ${p}_{2}\in \left(\frac{{p}^{2}}{{p}_{s}^{*}},p\right)$ such that

• (f3)

There exists $\alpha \in \left({p}_{1},p\right)$ such that

The first two results of our paper read as follows.

#### Theorem 1.1.

Let (f0)(f3) hold. Then there exists ${\lambda }_{\mathrm{0}}\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ such that problem () has at least a nontrivial weak solution ${u}_{\lambda }$ for all $\mathrm{0}\mathrm{<}\lambda \mathrm{<}{\lambda }_{\mathrm{0}}$.

Furthermore, if $f\left(x,t\right)$ is also odd in t, then for every $\lambda \in \left(0,+\mathrm{\infty }\right)$, problem () has infinitely many nontrivial weak solutions.

#### Theorem 1.2.

Let (f0)(f3) hold, and assume that $f\mathit{}\mathrm{\left(}x\mathrm{,}t\mathrm{\right)}$ is odd in t. Then, for every $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{+}\mathrm{\infty }\mathrm{\right)}$, problem () has a sequence of weak solutions $\mathrm{\left\{}{u}_{n}^{\lambda }\mathrm{\right\}}$ satisfying ${\mathrm{\parallel }{u}_{n}^{\lambda }\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$.

#### Remark 1.3.

Assumptions similar to (f1)(f3) were used in [22], where similar results were obtained for elliptic equation problems on a bounded domain. Condition (f3) is a local version of the subquadratic condition.

#### Remark 1.4.

In our assumptions ${p}_{1},{p}_{2}>\frac{{p}^{2}}{{p}_{s}^{*}}$ will be just used to give an ${L}^{\mathrm{\infty }}$-estimate for the solutions. Based on the proof of [12, Theorem 3.1], we can get a more suitable ${L}^{\mathrm{\infty }}$-estimate to our problem (). As far as we know (see [12]), similar bounds were obtained before only in some special cases, namely, for a semilinear fractional Laplacian equation with the reaction term independent of u, and for the eigenvalue problem of some fractional elliptic operators.

Secondly, we consider () with p-superlinear nonlinearity. In particular, we make the following assumptions on f just near origin:

• (f1’)

There exists ${q}_{1}\in \left(p,\frac{2p{p}_{s}^{*}}{p+{p}_{s}^{*}}\right)$ such that

• (f2’)

There exists ${q}_{2}\in \left(p,p+\gamma \right)$ such that

• (f3’)

There exists $\varpi \in \left(p,{p}_{s}^{*}\right)$ such that

#### Theorem 1.5.

Let (f0), (f1’)(f3’) hold. Then there exists ${\mathrm{\Lambda }}_{\mathrm{0}}\mathrm{>}\mathrm{1}$ such that problem () has at least a nontrivial weak solution ${u}_{\lambda }$ for all $\lambda \mathrm{>}{\mathrm{\Lambda }}_{\mathrm{0}}$.

#### Remark 1.6.

In our assumptions ${q}_{1}<\frac{2p{p}_{s}^{*}}{p+{p}_{s}^{*}}$, ${q}_{2} will also just be used to give an ${L}^{\mathrm{\infty }}$-estimate for the solutions.

We will prove our results via a variational approach following the methods of [22, 6]. The strategy is to modify and extend f to an appropriate $\stackrel{~}{f}$, and to show for the associated modified functional the existence of solutions with bounded ${L}^{\mathrm{\infty }}$ norm, therefore to obtain solutions for the original problem (). So the ${L}^{\mathrm{\infty }}$-estimate of the solution is very important. However, there are no ${L}^{p}$-estimates for fractional Laplace problems as the classic Laplace problem. Recently, in [12], Iannizzotto et al. proved a priori ${L}^{\mathrm{\infty }}$ bounds on the weak solutions of problem (P${}_{1}$). By this estimate and the Sobolev embedding, we are able to get a more suitable estimate of ${L}^{\mathrm{\infty }}$ norm of solutions and avoid further restriction on the behavior of f at infinity.

Throughout the paper, we denote by C various positive constants, whose values are not essential to the problem, and may be different from line to line. We denote the usual norm of ${L}^{q}\left(\mathrm{\Omega }\right)$ by $\parallel \cdot {\parallel }_{q}$ for $1⩽q⩽\mathrm{\infty }$. Moreover, let $0 be real numbers, and the fractional critical exponent be defined as

The paper is organized as follows. In Section 2, we introduce some preliminary notions and notations, and set the functional framework of our problem. In Section 3, we will prove Theorem 1.1. Section 4 is devoted to the proof Theorem 1.2. The proof of Theorem 1.5 will be given in Section 5.

## 2 Preliminary

In this preliminary section, for the reader’s convenience, we collect some basic results that will be used in the forthcoming sections.

Firstly, we introduce a variational setting for problem (). The Gagliardo seminorm is defined, for all measurable function $u:{ℝ}^{N}↦ℝ$, by

${\left[u\right]}_{s,p}={\left({\int }_{{ℝ}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+sp}}𝑑x𝑑y\right)}^{\frac{1}{p}}.$

The fractional Sobolev space

is endowed with the norm

${\parallel u\parallel }_{s,p}={\left({\int }_{{ℝ}^{N}}{|u|}^{p}𝑑x+{\left[u\right]}_{s,p}\right)}^{\frac{1}{p}}.$

In this paper, we will work in the closed linear subspace

which can be equivalently renormed by setting $\parallel \cdot \parallel ={\left[\cdot \right]}_{s,p}$ (see [8, Theorem 7.1]). It is readily seen that $\left(X\left(\mathrm{\Omega }\right),\parallel \cdot \parallel \right)$ is a uniformly convex Banach space and the following Sobolev embedding theorem holds.

#### Lemma 2.1 ([8]).

The embedding $X\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{↪}{L}^{q}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ is continuous for all $q\mathrm{\in }\mathrm{\left[}\mathrm{1}\mathrm{,}{p}_{s}^{\mathrm{*}}\mathrm{\right]}$, and compact for $q\mathrm{\in }\mathrm{\left(}\mathrm{1}\mathrm{,}{p}_{s}^{\mathrm{*}}\mathrm{\right)}$.

In [12], the fractional p-Laplacian is redefined variationally as the nonlinear operator $A:X\left(\mathrm{\Omega }\right)↦X{\left(\mathrm{\Omega }\right)}^{*}$ defined, for all $u,v\in X\left(\mathrm{\Omega }\right)$, by

$〈A\left(u\right),v〉={\int }_{{ℝ}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+sp}}𝑑x𝑑y.$

A weak solution of problem () is a function of $u\in X\left(\mathrm{\Omega }\right)$ such that

Since $X\left(\mathrm{\Omega }\right)$ is uniformly convex, A satisfies the following compactness condition:

• (S)

If $\left\{{u}_{n}\right\}$ is a sequence in $X\left(\mathrm{\Omega }\right)$ such that ${u}_{n}⇀u\text{in}X\left(\mathrm{\Omega }\right)$ and $〈A\left({u}_{n}\right),{u}_{n}-u〉\to 0$, then ${u}_{n}\to u\text{in}X\left(\mathrm{\Omega }\right)$.

#### Definition 2.2.

Let X be a Banach space. We say that the functional $I\in {C}^{1}\left(X,ℝ\right)$ satisfies the Palais–Smale condition at the level $c\in ℝ$ ((PS)${}_{c}$ in short) if any sequence $\left\{{u}_{n}\right\}\subset X$, satisfying $I\left({u}_{n}\right)\to c$, ${I}^{\prime }\left({u}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$, has a convergent subsequence. The functional I satisfies the (PS) condition if it satisfies the (PS)${}_{c}$ condition for any $c\in ℝ$.

When f is odd in t, we need the following critical point theorem, see [22, Lemma 2.4].

#### Lemma 2.3.

Let X be a Banach space. Let also $I\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}X\mathrm{,}\mathrm{R}\mathrm{\right)}$, and assume that I satisfies the (PS) condition, and that is bounded from below and even, with $I\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$. If for any $k\mathrm{\in }\mathrm{N}$, there exist a k-dimensional subspace ${X}^{k}$ and ${\rho }_{k}\mathrm{>}\mathrm{0}$ such that

$\underset{{X}^{k}\cap {S}_{{\rho }_{k}}}{sup}I<0,$

where ${S}_{{\rho }_{k}}\mathrm{=}\mathrm{\left\{}u\mathrm{\in }X\mathrm{:}\mathrm{\parallel }u\mathrm{\parallel }\mathrm{=}{\rho }_{k}\mathrm{\right\}}$, then I has a sequence of critical values ${c}_{k}\mathrm{<}\mathrm{0}$ satisfying ${c}_{k}\mathrm{\to }\mathrm{0}$ as $k\mathrm{\to }\mathrm{\infty }$.

## 3 Proof of Theorem 1.1

In this section we will prove Theorem 1.1. Since (f1)(f3) give the behavior of f just in $\mathrm{\Omega }×\left(-\delta ,\delta \right)$, the functional ${\int }_{\mathrm{\Omega }}F\left(x,u\right)𝑑x$ is not well-defined in $X\left(\mathrm{\Omega }\right)$. To overcome this difficulty, we need to modify and extend f to an appropriate $\stackrel{~}{f}$, in the spirit of the arguments developed in [22]. For this purpose, we first observe that (f1) and (f2) imply, for small $|t|$, that

(3.1)

Let $\rho \left(t\right)\in {C}^{1}\left(ℝ,\left[0,1\right]\right)$ be an even cut-off function verifying $t{\rho }^{\prime }\left(t\right)⩽0$ and

(3.2)

where $0<\tau <\frac{\delta }{2}$ is chosen such that (3.1), (3.2) and (f3) hold for $|s|⩽2\tau$. Set

$\stackrel{~}{F}\left(x,t\right)=\rho \left(t\right)F\left(x,t\right)+\left(1-\rho \left(t\right)\right){|t|}^{{p}_{1}},\stackrel{~}{f}\left(x,t\right)=\frac{\partial }{\partial t}\stackrel{~}{F}\left(x,t\right).$

It is easy to check that the following properties on $\stackrel{~}{f}$ hold.

#### Lemma 3.1.

Let (f0), (f1)(f3) be satisfied. Then the following hold:

• (i)

There exists a constant $C>0$ such that

• (ii)

We have

• (iii)

$\stackrel{~}{f}\left(x,t\right)$ is odd in t for all $t\in ℝ$ , if $f\left(x,t\right)$ is odd in t for $t\in \left(-\delta ,\delta \right)$.

#### Proof.

By the chosen of ρ, (i) and (iii) are simple. We will just show (ii) for ${p}_{1}⩽\alpha .

On the one hand, for $0⩽|t|⩽\tau$ and $|t|⩾2\tau$, we have

$\stackrel{~}{f}\left(x,t\right)t=\rho \left(t\right)tf\left(x,t\right)+{p}_{1}\left(1-\rho \left(t\right)\right){|t|}^{{p}_{1}}-{\rho }^{\prime }\left(t\right)t\left({|t|}^{{p}_{1}}-F\left(x,t\right)\right),$$\alpha \stackrel{~}{F}\left(x,t\right)-\stackrel{~}{f}\left(x,t\right)t=\rho \left(t\right)\left(\alpha F\left(x,t\right)-tf\left(x,t\right)\right)+\left(\alpha -{p}_{1}\right)\left(1-\rho \left(t\right)\right){|t|}^{{p}_{1}}+{\rho }^{\prime }\left(t\right)t\left({|t|}^{{p}_{1}}-F\left(x,t\right)\right),$

and the conclusion follows. On the other hand, by (f1), $F\left(x,t\right)⩾{|t|}^{{p}_{1}}$ for $\tau ⩽|t|⩽2\tau$. Using the fact that ${\rho }^{\prime }\left(t\right)t⩽0$ and $\alpha ⩾{p}_{1}$, by (f3), we get the conclusion. ∎

By Lemma 3.1, we can modify and extend f to get $\stackrel{~}{f}\in C\left(\mathrm{\Omega }×ℝ,ℝ\right)$ satisfying all properties listed in Lemma 3.1. Now we define

(3.3)

Then ${\stackrel{~}{I}}_{\lambda }\left(u\right)\in {C}^{1}\left(X\left(\mathrm{\Omega }\right),ℝ\right)$. We remark that $\stackrel{~}{f}\left(x,t\right)=f\left(x,t\right)$ for $\left(x,t\right)\in \mathrm{\Omega }×\left[-\tau ,\tau \right]$, and that a critical point u of ${\stackrel{~}{I}}_{\lambda }$ is a solution of () if and only if ${\parallel u\parallel }_{\mathrm{\infty }}⩽\tau$.

Now we investigate the properties of the functional ${\stackrel{~}{I}}_{\lambda }$.

#### Lemma 3.2.

${\stackrel{~}{I}}_{\lambda }\left(u\right)=〈{\stackrel{~}{I}}_{\lambda }^{\prime }\left(u\right),u〉=0$ if and only if $u\mathrm{=}\mathrm{0}$.

#### Proof.

By (3.3), it is easy to see that

$0={\stackrel{~}{I}}_{\lambda }\left(u\right)-\frac{1}{\alpha }〈{\stackrel{~}{I}}_{\lambda }^{\prime }\left(u\right),u〉=\left(\frac{1}{p}-\frac{1}{\alpha }\right){\parallel u\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}\left(\stackrel{~}{F}\left(x,u\right)-\frac{1}{\alpha }\stackrel{~}{f}\left(x,u\right)u\right)𝑑x.$

Since ${p}_{1}⩽\alpha and $\stackrel{~}{F}\left(x,u\right)-\frac{1}{\alpha }\stackrel{~}{f}\left(x,u\right)u⩾0$ for $u\in ℝ$, by Lemma 3.1, we get $u=0$. ∎

Lemma 3.2 implies that the trivial solution 0 of () is the unique critical point of ${\stackrel{~}{I}}_{\lambda }$ at the level 0.

Next we check that ${\stackrel{~}{I}}_{\lambda }\left(u\right)$ is coercive, i.e., ${\stackrel{~}{I}}_{\lambda }\left(u\right)\to \mathrm{\infty }$ as $\parallel u\parallel \to \mathrm{\infty }$, and ${\stackrel{~}{I}}_{\lambda }$ satisfies the (PS) condition.

#### Lemma 3.3.

The functional ${\stackrel{\mathrm{~}}{I}}_{\lambda }$ is bounded from below and satisfies the (PS) condition.

#### Proof.

From Lemma 3.1, we have

where C is a positive constant. Therefore,

${\stackrel{~}{I}}_{\lambda }\left(u\right)=\frac{1}{p}{\parallel u\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}\stackrel{~}{F}\left(x,u\right)𝑑x$$⩾\frac{1}{p}{\parallel u\parallel }^{p}-\lambda C{\int }_{\mathrm{\Omega }}\left({|u|}^{{p}_{1}}+{|u|}^{{p}_{2}}\right)𝑑x$$⩾\frac{1}{p}{\parallel u\parallel }^{p}-\lambda C\left({\parallel u\parallel }^{{p}_{1}}+{\parallel u\parallel }^{{p}_{2}}\right).$

Since ${p}_{2}<{p}_{1}, it follows that

(3.4)

that is, ${\stackrel{~}{I}}_{\lambda }\left(u\right)$ is coercive, and then is bounded from below.

Now we prove that ${\stackrel{~}{I}}_{\lambda }$ satisfies the (PS) condition. Let $\left\{{u}_{n}\right\}\subset X\left(\mathrm{\Omega }\right)$ be a (PS) sequence. Then there exists $M>0$, such that

(3.5)

By (3.4), it follows that $\left\{{u}_{n}\right\}$ is bounded in $X\left(\mathrm{\Omega }\right)$. By Lemma 2.1, we can assume that, up to a subsequence, for some $u\in X\left(\mathrm{\Omega }\right)$

${u}_{n}⇀u$${u}_{n}\to u$${u}_{n}\left(x\right)\to u\left(x\right)$

It follows, from (3.5) and the fact that ${u}_{n}⇀u$ in $X\left(\mathrm{\Omega }\right)$, that

As ${p}_{1},{p}_{2}\in \left(1,{p}_{s}^{*}\right)$, by Lemma 3.1 (i) and Hölder’s inequality, we have

$|{\int }_{\mathrm{\Omega }}\stackrel{~}{f}\left(x,{u}_{n}\right)\left({u}_{n}-u\right)𝑑x|⩽{\int }_{\mathrm{\Omega }}|\stackrel{~}{f}\left(x,{u}_{n}\right)||{u}_{n}-u|𝑑x$$⩽C{\int }_{\mathrm{\Omega }}\left({|{u}_{n}|}^{{p}_{1}-1}+{|{u}_{n}|}^{{p}_{2}-1}\right)|{u}_{n}-u|𝑑x$

Then, for any fixed $\lambda >0$,

Therefore, by condition (S), $\parallel {u}_{n}-u\parallel \to 0$ as $n\to \mathrm{\infty }$, and the functional ${\stackrel{~}{I}}_{\lambda }$ satisfies the (PS) condition. ∎

Now we are in the position to give the proof of Theorem 1.1.

#### Proof of Theorem 1.1.

By Lemma 3.3, for each $\lambda >0$, there exists some ${u}_{\lambda }\in X\left(\mathrm{\Omega }\right)$ such that

${\stackrel{~}{I}}_{\lambda }\left({u}_{\lambda }\right)=\underset{v\in X\left(\mathrm{\Omega }\right)}{inf}{\stackrel{~}{I}}_{\lambda }\left(v\right).$

We have ${u}_{\lambda }\ne 0$, since the trivial solution 0 is not a local minimizer. By Lemmas 2.1 and 3.1, and the fact that $〈{\stackrel{~}{I}}_{\lambda }^{\prime }\left({u}_{\lambda }\right),{u}_{\lambda }〉=0$, it follows that for some $C>0$,

${\parallel {u}_{\lambda }\parallel }^{p}=\lambda {\int }_{\mathrm{\Omega }}\stackrel{~}{f}\left(x,{u}_{\lambda }\right){u}_{\lambda }𝑑x⩽\lambda C\left({\parallel {u}_{\lambda }\parallel }^{{p}_{1}}+{\parallel {u}_{\lambda }\parallel }^{{p}_{2}}\right).$(3.6)

We claim that $\parallel {u}_{\lambda }\parallel ⩽1$ uniformly for $\lambda >0$ small. Otherwise, we have a sequence of ${\lambda }_{n}\to 0$ such that $\parallel {u}_{{\lambda }_{n}}\parallel >1$. Thus, ${\parallel {u}_{{\lambda }_{n}}\parallel }^{{p}_{2}}⩽{\parallel {u}_{{\lambda }_{n}}\parallel }^{{p}_{1}}$ since ${p}_{2}<{p}_{1}. By (3.6), we obtain

a contradiction with the assumption $\parallel {u}_{{\lambda }_{n}}\parallel >1$. Now, it follows from (3.6) that

Next we show that there exists ${\lambda }_{0}>0$ small enough such that ${\parallel {u}_{\lambda }\parallel }_{\mathrm{\infty }}⩽\tau$ for $\lambda \in \left(0,{\lambda }_{0}\right)$. We modify the proof of [12, Theorem 3.1]. By Lemma 3.1 (i) and (f2), we can easily get that

$|\stackrel{~}{f}\left(x,t\right)|⩽C\left({|t|}^{{p}_{2}-1}+{|t|}^{p-1}\right),\left(x,t\right)\in \mathrm{\Omega }×ℝ, 1+\frac{{p}_{2}}{p}>1+\frac{p}{{p}_{s}^{*}}.$

Then all the conditions of [12, Theorem 3.1] hold. Fix a weak solution ${u}_{\lambda }\in X\left(\mathrm{\Omega }\right)$, with ${u}_{\lambda }^{+}\ne 0$, of the following problem:

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}u& =\lambda \stackrel{~}{f}\left(x,u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill u\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega }.\end{array}$

We choose $\rho ⩾\mathrm{max}\left\{1,{\parallel {u}_{\lambda }\parallel }_{p}^{-1}\right\}$, set ${v}_{\lambda }={\left(\rho {\parallel {u}_{\lambda }\parallel }_{p}\right)}^{-1}{u}_{\lambda }$, so ${v}_{\lambda }\in X\left(\mathrm{\Omega }\right),{\parallel {v}_{\lambda }\parallel }_{p}={\rho }^{-1}$, and ${v}_{\lambda }$ is a weak solution of the auxiliary problem

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}{v}_{\lambda }& =\lambda {\left(\rho {\parallel {u}_{\lambda }\parallel }_{p}\right)}^{1-p}\stackrel{~}{f}\left(x,\rho {\parallel {u}_{\lambda }\parallel }_{p}{v}_{\lambda }\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill {v}_{\lambda }\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega }.\end{array}$(3.7)

For all $n\in ℕ$, we set ${v}_{n,\lambda }={\left({v}_{\lambda }-\tau +\tau {2}^{-n}\right)}^{+}$. Then ${v}_{n,\lambda }\in X\left(\mathrm{\Omega }\right),{v}_{0,\lambda }={v}_{\lambda }^{+}$, and for all $n\in ℕ$, we have that $0⩽{v}_{n+1,\lambda }\left(x\right)⩽{v}_{n,\lambda }\left(x\right)$ and ${v}_{n,\lambda }\left(x\right)\to {\left({v}_{\lambda }\left(x\right)-\tau \right)}^{+}$ for a.e. $x\in \mathrm{\Omega }$ as $n\to \mathrm{\infty }$. Moreover, the following inclusion holds (up to a Lebesgue null set):

$\left\{{v}_{n+1,\lambda }>0\right\}\subseteq \left\{0<{v}_{\lambda }<\left({2}^{n+1}-1\right){v}_{n,\lambda }\right\}\cap \left\{{v}_{n,\lambda }>\tau {2}^{-n-1}\right\}.$(3.8)

For all $n\in ℕ$, we set ${R}_{n,\lambda }={\parallel {v}_{n,\lambda }\parallel }_{p}^{p}$, so ${R}_{0,\lambda }={\parallel {v}_{\lambda }^{+}\parallel }_{p}^{p}⩽{\rho }^{-p}$, and $\left\{{R}_{n,\lambda }\right\}$ is a nonincreasing sequence in $\left[0,1\right]$. We shall prove that ${R}_{n,\lambda }\to 0$ as $n\to \mathrm{\infty }$. By Hölder’s inequality, the fractional Sobolev inequality (see [8, Theorem 6.5]), (3.8), and Chebyshev’s inequality, for all $n\in ℕ$, we have

${R}_{n+1,\lambda }⩽{|\left\{{v}_{n+1,\lambda }>0\right\}|}^{1-\frac{p}{{p}_{s}^{*}}}\parallel {v}_{n+1,\lambda }{\parallel }_{{p}_{s}^{*}}^{p}$$⩽C{|\left\{{v}_{n,\lambda }>\tau {2}^{-n-1}\right\}|}^{1-\frac{p}{{p}_{s}^{*}}}\parallel {v}_{n+1,\lambda }{\parallel }^{p}$$⩽C{\tau }^{p\left(\frac{p}{{p}_{s}^{*}}-1\right)}{2}^{\left(p-\frac{{p}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{R}_{n,\lambda }^{1-\frac{p}{{p}_{s}^{*}}}{\parallel {v}_{n+1,\lambda }\parallel }^{p}$$⩽{C}^{\prime }{2}^{\left(p-\frac{{p}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{R}_{n,\lambda }^{1-\frac{p}{{p}_{s}^{*}}}{\parallel {v}_{n+1,\lambda }\parallel }^{p}.$(3.9)

So, what we need now is an estimate of $\parallel {v}_{n+1,\lambda }\parallel$. Using the elementary inequality

${|{\zeta }^{+}-{\eta }^{+}|}^{p}⩽{|\zeta -\eta |}^{p-2}\left(\zeta -\eta \right)\left({\zeta }^{+}-{\eta }^{+}\right),\zeta ,\eta \in ℝ,$

testing (3.7) with ${v}_{n+1,\lambda }$, and applying (3.8), for any fixed $\lambda \in \left(0,1\right)$, we obtain

${\parallel {v}_{n+1,\lambda }\parallel }^{p}⩽〈A\left({v}_{\lambda }\right),{v}_{n+1,\lambda }〉$$⩽\lambda {\int }_{\mathrm{\Omega }}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{p}\right)}^{1-p}\stackrel{~}{f}\left(x,\rho {\parallel {u}_{\lambda }\parallel }_{p}{v}_{\lambda }\right){v}_{n+1,\lambda }𝑑x$$⩽C{\int }_{\left\{{v}_{n+1,\lambda }>0\right\}}\left({\left(\rho {\parallel {u}_{\lambda }\parallel }_{p}\right)}^{{p}_{2}-p}{|{v}_{\lambda }|}^{{p}_{2}-1}+{|{v}_{\lambda }|}^{p-1}\right){v}_{n+1,\lambda }𝑑x$$⩽C{\int }_{\left\{{v}_{n+1,\lambda }>0\right\}}\left({\left({2}^{n+1}-1\right)}^{{p}_{2}-1}{|{v}_{n,\lambda }|}^{{p}_{2}}+{\left({2}^{n+1}-1\right)}^{p-1}{|{v}_{n,\lambda }|}^{p}\right)𝑑x$$⩽C{2}^{\left(n+1\right)\left(p-1\right)}{R}_{n,\lambda }^{\frac{{p}_{2}}{p}}.$(3.10)

Combining (3.9) with (3.10), we have

${R}_{n+1,\lambda }⩽C{2}^{\left(2p-1-\frac{{p}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{R}_{n,\lambda }^{1+\frac{{p}_{2}}{p}-\frac{p}{{p}_{s}^{*}}}=C{2}^{\left(2p-1-\frac{{p}^{2}}{{p}_{s}^{*}}\right)}{H}^{n}{R}_{n,\lambda }^{1+\beta }⩽{H}^{n}{\left({C}_{0}{R}_{n,\lambda }\right)}^{1+\beta },$(3.11)

where

Similar to [12], provided that

$\rho =\mathrm{max}\left\{{\left(\frac{{C}_{0}^{1+\beta }}{\nu }\right)}^{\frac{1}{p\beta }},\frac{1}{{\parallel {u}_{\lambda }\parallel }_{p}}\right\}$

is large enough, we can prove, for all $n\in ℕ$, that

(3.12)

We argue by induction. We already know that ${R}_{0}⩽\frac{1}{{\rho }^{p}}$. Assuming that (3.12) holds for some $n\in ℕ$, by (3.11), we have

${R}_{n+1,\lambda }⩽{H}^{n}{\left({C}_{0}{R}_{n,\lambda }\right)}^{1+\beta }⩽{H}^{n}{C}_{0}^{1+\beta }{\left(\frac{{\nu }^{n}}{{\rho }^{p}}\right)}^{1+\beta }⩽\frac{{C}_{0}^{1+\beta }}{{\rho }^{p\beta }}\frac{{\nu }^{n}}{{\rho }^{p}}⩽\frac{{\nu }^{n+1}}{{\rho }^{p}}.$

By (3.12), we have ${R}_{n,\lambda }\to 0$. This implies that ${v}_{n,\lambda }\left(x\right)\to 0$ as $n\to \mathrm{\infty }$ for a.e. $x\in \mathrm{\Omega }$. So, ${v}_{\lambda }\left(x\right)⩽\tau$ for a.e. $x\in \mathrm{\Omega }$. An analogous argument applies to $-{v}_{\lambda }$, so we have ${\parallel {v}_{\lambda }\parallel }_{\mathrm{\infty }}⩽\tau$, hence ${u}_{\lambda }\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, and by the fractional Sobolev embedding,

That is to say, we can find ${\lambda }_{0}>0$ such that ${\parallel {u}_{\lambda }\parallel }_{\mathrm{\infty }}⩽\tau$ for $0<\lambda <{\lambda }_{0}$. Hence, ${u}_{\lambda }$ is a nontrivial solution of the original problem (). The proof is complete. ∎

## 4 Proof of Theorem 1.2

In this section, we deal with the case where f is odd near the origin. First of all, for any fixed $\lambda >0$, we will get the existence of the infinitely many critical points of the functional ${\stackrel{~}{I}}_{\lambda }$, by Lemma 2.3. By Lemmas 3.1 and 3.3, and (f2), we have in hand the facts that ${\stackrel{~}{I}}_{\lambda }$ satisfies the (PS) condition, and that is even and bounded from below, with ${\stackrel{~}{I}}_{\lambda }\left(0\right)=0$. Hence, it suffices to find, for any $k\in ℕ$, a subspace ${X}^{k}$ and ${\rho }_{k}>0$ such that

$\underset{u\in {X}^{k}\bigcap {S}_{{\rho }_{k}}}{sup}{\stackrel{~}{I}}_{\lambda }\left(u\right)<0.$(4.1)

For any $k\in ℕ$, we find k independent functions ${\phi }_{i}\in X\left(\mathrm{\Omega }\right)$, $i=1,\mathrm{\dots },k$, and set ${X}^{k}:=\mathrm{span}\left\{{\phi }_{1},{\phi }_{2},\mathrm{\dots },{\phi }_{k}\right\}$. By (f1) and the definition of $\stackrel{~}{F}\left(x,t\right)$, we have $\stackrel{~}{F}\left(x,t\right)⩾C{|t|}^{{p}_{1}}$ for $t\in ℝ$. Then

${\stackrel{~}{I}}_{\lambda }\left(u\right)=\frac{1}{p}\parallel u{\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}\stackrel{~}{F}\left(x,u\right)dx⩽\frac{1}{p}\parallel u{\parallel }^{p}-\lambda C{\int }_{\mathrm{\Omega }}|u{|}^{{p}_{1}}dx=\frac{1}{p}\parallel u{\parallel }^{p}-\lambda C\parallel u{\parallel }_{{p}_{1}}^{{p}_{1}}.$

Since all norms on ${X}^{k}$ are equivalent, ${p}_{1}, by choosing ${\rho }_{k}>0$ small enough, (4.1) holds. With all conditions of Lemma 2.3 being verified, we get a sequence of critical points ${u}_{n}^{\lambda }\in X\left(\mathrm{\Omega }\right)$ with ${\stackrel{~}{I}}_{\lambda }\left({u}_{n}^{\lambda }\right)={c}_{n}^{\lambda }\to 0$ and ${c}_{n}^{\lambda }<0$ as $n\to \mathrm{\infty }$. In particular, $\left\{{u}_{n}^{\lambda }\right\}$ is a (PS)${}_{0}$ sequence of ${\stackrel{~}{I}}_{\lambda }$ and has a convergent subsequence, still denoted by $\left\{{u}_{n}^{\lambda }\right\}$. By Lemma 3.2,

Next, following similar arguments as in the proof of Theorem 1.1, we can choose n large enough such that $\rho =1/{\parallel {u}_{n}^{\lambda }\parallel }_{p}$. Then we also get ${\parallel {u}_{n}^{\lambda }\parallel }_{\mathrm{\infty }}⩽\tau$ for n large enough. That is to say, for n large enough, ${u}_{n}^{\lambda }$ are solutions of the original problem (). The proof is complete.∎

## 5 Proof of Theorem 1.5

In this section, we deal with the superlinear case and give the proof of Theorem 1.5. We need to modify the nonlinearity as before. Similarly, we first observe that (f1’) and (f2’) imply that for $|t|$ small enough,

(5.1)

and

(5.2)

Let $\rho \left(s\right)$ be the cut-off function defined in (3.1), where $0<\tau <\frac{\delta }{2}$ is chosen such that (5.1), (5.2) and (f3’) hold for $|t|⩽2\tau$. Set

$\overline{F}\left(x,t\right)=\rho \left(t\right)F\left(x,t\right)+\left(1-\rho \left(t\right)\right){|t|}^{{q}_{1}},\overline{f}\left(x,t\right)=\frac{\partial }{\partial t}\overline{F}\left(x,t\right).$

As in Lemma 3.1, it is easy to check that $\overline{f}$ has the following properties.

#### Lemma 5.1.

• (i)

There exists $C>0$ such that

• (ii)

We have that

(5.3)

where $\theta =\mathrm{min}\left({q}_{1},\varpi \right)$.

We introduce the functional

${\overline{I}}_{\lambda }\left(u\right)=\frac{1}{p}{\parallel u\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}\overline{F}\left(x,u\right)𝑑x,$(5.4)

which is well defined on $X\left(\mathrm{\Omega }\right)$, by Lemma 5.1.

#### Lemma 5.2.

The functional ${\overline{I}}_{\lambda }$ satisfies the (PS) condition.

#### Proof.

Let $\left\{{u}_{n}\right\}$ be a (PS) sequence for ${\overline{I}}_{\lambda }$, that is,

From (5.3), we observe that since $\theta >p$, for n large enough,

$M+1+\parallel {u}_{n}\parallel ⩾{\overline{I}}_{\lambda }\left({u}_{n}\right)-\frac{1}{\theta }〈{\overline{I}}_{\lambda }^{\prime }\left({u}_{n}\right),{u}_{n}〉⩾\left(\frac{1}{p}-\frac{1}{\theta }\right){\parallel {u}_{n}\parallel }^{p}.$(5.5)

Hence, $\left\{{u}_{n}\right\}$ is bounded, and the rest of proof is standard as in Lemma 3.3. ∎

The functional ${\overline{I}}_{\lambda }$ enjoys the mountain pass geometry.

#### Lemma 5.3.

• (i)

There exist ${\rho }_{\lambda }=\rho \left(\lambda \right)>0$ and ${\beta }_{\lambda }=\beta \left({\rho }_{\lambda },\lambda \right)>0$ such that

• (ii)

Let $e\in X\left(\mathrm{\Omega }\right)$ with $0<{\parallel e\parallel }_{\mathrm{\infty }}<2\tau$ . Then there exists ${\mathrm{\Lambda }}_{0}>0$ such that $\parallel e\parallel >{\rho }_{\lambda }$ and ${\overline{I}}_{\lambda }\left(e\right)<0$ for $\lambda >{\mathrm{\Lambda }}_{0}$.

#### Proof.

By Lemma 5.1 and (5.4), we see that

${\overline{I}}_{\lambda }\left(u\right)⩾\frac{1}{p}{\parallel u\parallel }^{p}-C\lambda {|u|}_{{q}_{1}}^{{q}_{1}}⩾{\parallel u\parallel }^{p}\left(\frac{1}{p}-C\lambda {\parallel u\parallel }^{{q}_{1}-p}\right),$

where C is independent of λ. Taking ${\rho }_{\lambda }={\left(2pC\lambda \right)}^{\frac{1}{p-{q}_{1}}}$, we have

We also see that $\parallel e\parallel >{\rho }_{\lambda }$ for λ large enough, since ${\rho }_{\lambda }\to 0$ as $\lambda \to \mathrm{\infty }$. Moreover, we have

${\overline{I}}_{\lambda }\left(e\right)=\frac{1}{p}{\parallel e\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}\overline{F}\left(x,e\right)𝑑x.$

Hence, there exists ${\mathrm{\Lambda }}_{0}>0$ such that ${\overline{I}}_{\lambda }\left(e\right)<0$ for $\lambda >{\mathrm{\Lambda }}_{0}$. ∎

From Lemmas 5.2 and 5.3, and the mountain pass theorem, we see that ${\overline{I}}_{\lambda }$ has a critical value ${c}_{\lambda }$ and

${c}_{\lambda }=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{t\in \left[0,1\right]}{\mathrm{max}}{\overline{I}}_{\lambda }\left(\gamma \left(t\right)\right),$

where $\mathrm{\Gamma }=\left\{\gamma \in C\left(\left[0,1\right],E\right):\gamma \left(0\right)=0,\gamma \left(1\right)=e\right\}$.

In order to get the estimate of the critical level ${c}_{\lambda }$, we introduce the following functional:

${J}_{\lambda }\left(u\right)=\frac{1}{p}{\parallel u\parallel }^{p}-\lambda {\int }_{\mathrm{\Omega }}{|u|}^{{q}_{2}}𝑑x.$(5.6)

#### Lemma 5.4.

There exists $C\mathrm{>}\mathrm{0}$ independent of $\lambda \mathrm{>}\mathrm{1}$ such that

${c}_{\lambda }⩽C{\lambda }^{-\frac{p}{{q}_{2}-p}}.$

#### Proof.

By direct calculations, from (5.6), we obtain that

$\underset{t⩾0}{\mathrm{max}}{J}_{\lambda }\left(te\right)=\frac{{q}_{2}-p}{p{q}_{2}}{\left(\frac{1}{\lambda {q}_{2}}\right)}^{\frac{p}{{q}_{2}-p}}{\left(\frac{{\parallel e\parallel }^{p}}{{\parallel e\parallel }_{{q}_{2}}^{{q}_{2}}}\right)}^{\frac{p}{{q}_{2}-p}}.$

Since ${\parallel te\parallel }_{\mathrm{\infty }}<2\tau$, $t\in \left[0,1\right]$, and (5.1) holds, there exists $C>0$ independent of $\lambda >1$ such that

${c}_{\lambda }⩽\underset{t\in \left[0,1\right]}{\mathrm{max}}{\overline{I}}_{\lambda }\left(te\right)⩽\underset{t\in \left[0,1\right]}{\mathrm{max}}{J}_{\lambda }\left(te\right)⩽C{\lambda }^{-\frac{p}{{q}_{2}-p}}.\mathit{∎}$

Now we give the proof of Theorem 1.5.

#### Proof of Theorem 1.5.

Let ${u}_{\lambda }$ be a critical point of ${\overline{I}}_{\lambda }$ with critical value ${c}_{\lambda }$. Similar to (5.5), by Lemma 5.4, we have

Next, similar to Theorem 1.1, we will prove that for λ large enough, the critical point ${u}_{\lambda }$ is also the solution of the original problem (). By Lemma 5.1 (i) and (f1’), we can easily get that

$|\overline{f}\left(x,t\right)|⩽C\left({|t|}^{{q}_{1}-1}+{|t|}^{p-1}\right),\left(x,t\right)\in \mathrm{\Omega }×ℝ, 2>\frac{{q}_{1}}{p}+\frac{{q}_{1}}{{p}_{s}^{*}}.$

Then all the conditions of [12, Theorem 3.1] hold. Fix a weak solution ${u}_{\lambda }\in X\left(\mathrm{\Omega }\right)$, with ${u}_{\lambda }^{+}\ne 0$, of the following problem:

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}u& =\lambda \overline{f}\left(x,u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill u\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega }.\end{array}$

We choose $\rho ⩾\mathrm{max}\left\{1,{\parallel {u}_{\lambda }\parallel }_{{q}_{1}}^{-1}\right\}$ and set ${v}_{\lambda }={\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{-1}{u}_{\lambda }$. Then ${v}_{\lambda }\in X\left(\mathrm{\Omega }\right),{\parallel {v}_{\lambda }\parallel }_{{q}_{1}}={\rho }^{-1}$ and ${v}_{\lambda }$ is a weak solution of the auxiliary problem

$\left\{\begin{array}{cccc}\hfill {\left(-\mathrm{\Delta }\right)}_{p}^{s}{v}_{\lambda }& =\lambda {\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{1-p}\overline{f}\left(x,\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}{v}_{\lambda }\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill {v}_{\lambda }\left(x\right)& =0,\hfill & & \hfill x\in {ℝ}^{N}\setminus \mathrm{\Omega }.\end{array}$(5.7)

For all $n\in ℕ$, we set ${v}_{n,\lambda }={\left({v}_{\lambda }-\tau +\tau {2}^{-n}\right)}^{+}$. Then ${v}_{n,\lambda }\in X\left(\mathrm{\Omega }\right)$, ${v}_{0,\lambda }={v}_{\lambda }^{+}$ and for all $n\in ℕ$, we have that $0⩽{v}_{n+1,\lambda }\left(x\right)⩽{v}_{n,\lambda }\left(x\right)$ and ${v}_{n,\lambda }\left(x\right)\to {\left({v}_{\lambda }\left(x\right)-\tau \right)}^{+}$ for a.e. $x\in \mathrm{\Omega }$ as $n\to \mathrm{\infty }$. Moreover, the following inclusion holds (up to a Lebesgue null set):

$\left\{{v}_{n+1,\lambda }>0\right\}\subseteq \left\{0<{v}_{\lambda }<\left({2}^{n+1}-1\right){v}_{n,\lambda }\right\}\cap \left\{{v}_{n,\lambda }>\tau {2}^{-n-1}\right\}.$(5.8)

For all $n\in ℕ$, we set ${R}_{n,\lambda }={\parallel {v}_{n,\lambda }\parallel }_{{q}_{1}}^{{q}_{1}}$. Then ${R}_{0,\lambda }={\parallel {v}_{\lambda }^{+}\parallel }_{{q}_{1}}^{{q}_{1}}⩽{\rho }^{-{q}_{1}}$ and $\left\{{R}_{n,\lambda }\right\}$ is a nonincreasing sequence in $\left[0,1\right]$. We shall prove that ${R}_{n,\lambda }\to 0$ as $n\to \mathrm{\infty }$. By Hölder’s inequality, the fractional Sobolev inequality (see [8, Theorem 6.5]), (5.8) and Chebyshev’s inequality, for all $n\in ℕ$, we have

${R}_{n+1,\lambda }⩽|\left\{{v}_{n+1,\lambda }>0\right\}{|}^{1-\frac{{q}_{1}}{{p}_{s}^{*}}}\parallel {v}_{n+1,\lambda }{\parallel }_{{p}_{s}^{*}}^{{q}_{1}}$$⩽C|\left\{{v}_{n,\lambda }>\tau {2}^{-n-1}\right\}{|}^{1-\frac{{q}_{1}}{{p}_{s}^{*}}}\parallel {v}_{n+1,\lambda }{\parallel }^{{q}_{1}}$$⩽C{\tau }^{{q}_{1}\left(\frac{{q}_{1}}{{p}_{s}^{*}}-1\right)}{2}^{\left({q}_{1}-\frac{{q}_{1}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{R}_{n,\lambda }^{1-\frac{{q}_{1}}{{p}_{s}^{*}}}{\parallel {v}_{n+1,\lambda }\parallel }^{{q}_{1}}$$⩽{C}^{\prime }{2}^{\left({q}_{1}-\frac{{q}_{1}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{R}_{n,\lambda }^{1-\frac{{q}_{1}}{{p}_{s}^{*}}}{\parallel {v}_{n+1,\lambda }\parallel }^{{q}_{1}}.$(5.9)

So, what we need now is an estimate of $\parallel {v}_{n+1,\lambda }\parallel$. Using the elementary inequality

${|{\zeta }^{+}-{\eta }^{+}|}^{p}⩽{|\zeta -\eta |}^{p-2}\left(\zeta -\eta \right)\left({\zeta }^{+}-{\eta }^{+}\right),\zeta ,\eta \in ℝ,$

testing (5.7) with ${v}_{n+1,\lambda }$, and applying (5.8), for any fixed large $\lambda >1$, we obtain

${\parallel {v}_{n+1,\lambda }\parallel }^{p}⩽〈A\left({v}_{\lambda }\right),{v}_{n+1,\lambda }〉$$⩽\lambda {\int }_{\mathrm{\Omega }}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{1-p}\overline{f}\left(x,\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}{v}_{\lambda }\right){v}_{n+1,\lambda }𝑑x$$⩽\lambda C{\int }_{\left\{{v}_{n+1,\lambda }>0\right\}}\left({\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{{q}_{1}-p}{|{v}_{\lambda }|}^{{q}_{1}-1}+{|{v}_{\lambda }|}^{p-1}\right){v}_{n+1,\lambda }𝑑x$$⩽\lambda C{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{{q}_{1}-p}{\int }_{\left\{{v}_{n+1,\lambda }>0\right\}}\left({\left({2}^{n+1}-1\right)}^{{q}_{1}-1}{|{v}_{n,\lambda }|}^{{q}_{1}}+{\left({2}^{n+1}-1\right)}^{p-1}{|{v}_{n,\lambda }|}^{p}\right)𝑑x$$⩽\lambda C{2}^{\left(n+1\right)\left({q}_{1}-1\right)}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{{q}_{1}-p}{R}_{n,\lambda }^{\frac{p}{{q}_{1}}}.$(5.10)

Combining (5.9) with (5.10), we have

${R}_{n+1,\lambda }⩽\lambda C{2}^{\left({q}_{1}+\frac{{q}_{1}^{2}}{p}-\frac{{q}_{1}}{p}-\frac{{q}_{1}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}{R}_{n,\lambda }^{2-\frac{{q}_{1}}{{p}_{s}^{*}}}$$=\lambda C{2}^{\left({q}_{1}+\frac{{q}_{1}^{2}}{p}-\frac{{q}_{1}}{p}-\frac{{q}_{1}^{2}}{{p}_{s}^{*}}\right)\left(n+1\right)}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}{H}^{n}{R}_{n,\lambda }^{1+\beta }$$⩽{H}^{n}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}\lambda {\left({C}_{0}{R}_{n,\lambda }\right)}^{1+\beta },$(5.11)

where

Similar to [12], provided that

$\rho =\mathrm{max}\left\{\frac{1}{{\parallel {u}_{\lambda }\parallel }_{{q}_{1}}},{\nu }^{-\frac{1}{\gamma }}{\left(\lambda {C}_{0}^{1+\beta }{\parallel {u}_{\lambda }\parallel }_{{q}_{1}}^{\left(\frac{{q}_{1}^{2}}{p}-{q}_{1}\right)}\right)}^{\frac{1}{\gamma }}\right\}$

is big enough, we can prove that for all $n\in ℕ$,

(5.12)

We argue by induction. We already know that ${R}_{0}⩽\frac{1}{{\rho }^{p}}$. Assuming that (5.12) holds for some $n\in ℕ$, by (5.11), we have

${R}_{n+1,\lambda }⩽{H}^{n}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}\lambda {\left({C}_{0}{R}_{n,\lambda }\right)}^{1+\beta }$$⩽{H}^{n}{\left(\rho {\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}\lambda C_{0}{}^{1+\beta }{\left(\frac{{\nu }^{n}}{{\rho }^{{q}_{1}}}\right)}^{1+\beta }$$⩽\frac{\lambda {\left({C}_{0}\right)}^{1+\beta }{\left({\parallel {u}_{\lambda }\parallel }_{{q}_{1}}\right)}^{\frac{{q}_{1}^{2}}{p}-{q}_{1}}}{{\rho }^{\gamma }}\frac{{\nu }^{n}}{{\rho }^{{q}_{1}}}$$⩽\frac{{\nu }^{n+1}}{{\rho }^{{q}_{1}}}.$

By (5.12), we have ${R}_{n,\lambda }\to 0$. This implies that ${v}_{n,\lambda }\left(x\right)\to 0$ as $n\to \mathrm{\infty }$ for a.e. $x\in \mathrm{\Omega }$. So ${v}_{\lambda }\left(x\right)⩽\tau$ for a.e. $x\in \mathrm{\Omega }$. An analogous argument applies to $-{v}_{\lambda }$, so we have ${\parallel {v}_{\lambda }\parallel }_{\mathrm{\infty }}<\tau$. Hence, ${u}_{\lambda }\in {L}^{\mathrm{\infty }}\left(\mathrm{\infty }\right)$, and by the fractional Sobolev embedding

since ${q}_{2}<\gamma +p$. That is to say, we can find ${\mathrm{\Lambda }}_{0}>0$ such that ${\parallel {u}_{\lambda }\parallel }_{\mathrm{\infty }}⩽\tau$ for $\lambda >{\mathrm{\Lambda }}_{0}$. Hence, ${u}_{\lambda }$ is a nontrivial solution of the original problem (). The proof is complete. ∎

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Accepted: 2016-09-11

Published Online: 2016-11-23

Published in Print: 2018-11-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11526126

Award identifier / Grant number: 11271264

Award identifier / Grant number: 11571209

This work was supported by NSFC grant no. 11526126, and was partly supported by NSFC grants nos. 11271264 and 11571209.

Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 4, Pages 485–496, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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