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Volume 9, Issue 1

# Noncoercive resonant (p,2)-equations with concave terms

Nikolaos S. Papageorgiou
/ Chao Zhang
• Corresponding author
• Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
• Email
• Other articles by this author:
Published Online: 2018-10-11 | DOI: https://doi.org/10.1515/anona-2018-0175

## Abstract

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a $\left(p,2\right)$-equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

MSC 2010: 35J20; 35J60; 58E05

## 1 Introduction

Let $\mathrm{\Omega }\subseteq {ℝ}^{N}$ be a bounded domain with a ${C}^{2}$-boundary $\partial \mathrm{\Omega }$. In this paper, we study the following parametric $\left(p,2\right)$-equation:

(1.1)

In this problem, for $r\in \left(1,+\mathrm{\infty }\right)$, by ${\mathrm{\Delta }}_{r}$ we denote the r-Laplace differential operator defined by

When $r=2$, we have the usual Laplacian ${\mathrm{\Delta }}_{2}=\mathrm{\Delta }$ defined by

So, in problem (1.1) the differential operator is the sum of a p-Laplacian and a Laplacian (a $\left(p,2\right)$-equation). Such an operator is nonhomogeneous and this is a source of difficulties in the analysis of problem (1.1). In the reaction (right-hand side of (1.1)), we have the competing effects of two nonlinearities. One is a concave term $u\to \lambda {|u|}^{q-2}u$ (recall $1) and the other is a Caratheodory perturbation $f\left(z,x\right)$ (that is, for all $x\in ℝ$, $z↦f\left(z,x\right)$ is measurable and for a.a. $z\in \mathrm{\Omega }$, $x↦f\left(z,x\right)$ is continuous). We assume that asymptotically as $x\to ±\mathrm{\infty }$, $f\left(z,\cdot \right)$ is resonant with respect to the principle eigenvalue ${\stackrel{^}{\lambda }}_{1}\left(p\right)>0$ of $\left(-{\mathrm{\Delta }}_{p},{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$. The resonance occurs from the right of the principal eigenvalue ${\stackrel{^}{\lambda }}_{1}\left(p\right)$ in the sense that

where $F\left(z,x\right)={\int }_{0}^{x}f\left(z,x\right)𝑑s$. This makes the energy functional of problem (1.1) indefinite. Our goal is to prove a multiplicity theorem for problem (1.1), providing sign information for all the solutions produced. We show that for all $\lambda >0$ small, problem (1.1) has at least six nontrivial smooth solutions, two positive, two negative and two nodal (that is, sign changing).

Our tools come from critical point theory, together with suitable truncation and comparison techniques and Morse theory (critical groups).

Boundary value problems driven by a combination of differential operators of different nature (such as $\left(p,2\right)$-equations) arise in the problems of mathematical physics. We mention the works of Benci, Fortunato and Pisani [3] concerning models of particle physics (existence of soliton-type solutions), Cherfils and Ilyason [5] on reaction-diffusion systems, and Wilhelmsson [29] on plasma physics. A survey of recent works on $\left(p,q\right)$-equations $\left(1 can be found in the paper of Marano and Mosconi [16]. The particular case of $\left(p,2\right)$-equations presents special interest and stronger results can be proved for such equations. In this direction we mention the works of Aizicovici, Papageorgiou and Staicu [2], Cingolani and Degiovanni [6], Papageorgiou and Radulescu [21, 22], Papageorgiou, Radulescu and Repovs [23, 24], Sun [27], and Sun, Zhang and Su [28]. From the aforementioned works, only [22] deals with equations which are resonant with respect to ${\stackrel{^}{\lambda }}_{1}\left(p\right)$ from the right (noncoercive problems), where the existence of two nontrivial solutions with no sign information is proved . Resonant $\left(p,2\right)$-equations with parametric concave terms were considered by Papageorgiou and Winkert [25]. However, in their equation, the concave term enters with a negative sign and this leads to a coercive energy functional.

## 2 Mathematical background and hypotheses

Let X be a Banach space and ${X}^{*}$ its topological dual. By $〈\cdot ,\cdot 〉$ we denote the duality brackets for the pair $\left({X}^{*},X\right)$. Given $\phi \in {C}^{1}\left(X,ℝ\right)$, we say that φ satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds: Every sequence ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq X$ such that ${\left\{\phi \left({u}_{n}\right)\right\}}_{n\ge 1}\subseteq ℝ$ is bounded and

This is a compactness-type condition on the functional φ, more general than the usual Palais–Smale condition. It leads to a deformation theorem from which one can deduce the minimax theory of the critical values of φ. One of the main results in that theory is the so-called “mountain pass theorem”, which we recall here.

#### Theorem 2.1.

If X is a Banach space, $\phi \mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}X\mathrm{,}\mathrm{R}\mathrm{\right)}$ satisfies the C-condition, ${u}_{\mathrm{0}}\mathrm{,}{u}_{\mathrm{1}}\mathrm{\in }X$, ${\mathrm{\parallel }{u}_{\mathrm{1}}\mathrm{-}{u}_{\mathrm{0}}\mathrm{\parallel }}_{X}\mathrm{>}\rho \mathrm{>}\mathrm{0}$,

$\mathrm{max}\left\{\phi \left({u}_{0}\right),\phi \left({u}_{1}\right)\right\}

and

then $c\mathrm{\ge }{\eta }_{\rho }$ and c is a critical value of φ.

In the analysis of problem (1.1), we will use the Sobolev spaces ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ and ${H}_{0}^{1}\left(\mathrm{\Omega }\right)$. Since $2, we have that ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)↪{H}_{0}^{1}\left(\mathrm{\Omega }\right)$ densely. By $\parallel \cdot \parallel$ we denote the norm of ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. By the Poincaré inequality, we have

Also we will use the Banach space ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)=\left\{u\in {C}^{1}\left(\overline{\mathrm{\Omega }}\right):{u|}_{\partial \mathrm{\Omega }}=0\right\}$. This is an ordered Banach space with positive (order) cone

This cone has a nonempty interior given by

Here $\frac{\partial u}{\partial n}={\left(Du,n\right)}_{{ℝ}^{N}}$, with $n\left(\cdot \right)$ being the outward unit normal on $\partial \mathrm{\Omega }$.

For every $r\in \left(1,+\mathrm{\infty }\right)$, let ${A}_{r}:{W}_{0}^{1,r}\left(\mathrm{\Omega }\right)\to {W}^{-1,{r}^{\prime }}\left(\mathrm{\Omega }\right)={W}_{0}^{1,r}{\left(\mathrm{\Omega }\right)}^{*}$ ($\frac{1}{r}+\frac{1}{{r}^{\prime }}=1$) be the map defined by

For this map, we have the following result (see [18, p. 40]).

#### Proposition 2.2.

The map ${A}_{r}\mathit{}\mathrm{\left(}\mathrm{\cdot }\mathrm{\right)}$ is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone, too) and of type ${\mathrm{\left(}S\mathrm{\right)}}_{\mathrm{+}}$, that is, if ${u}_{n}\stackrel{𝑤}{\mathrm{\to }}u$ in ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}r}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and

$\underset{n\to \mathrm{\infty }}{lim sup}〈{A}_{r}\left({u}_{n}\right),{u}_{n}-u〉\le 0,$

then ${u}_{n}\mathrm{\to }u$ in ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}r}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$. If $r\mathrm{=}\mathrm{2}$, then ${A}_{\mathrm{2}}\mathrm{=}A\mathrm{\in }\mathcal{L}\mathit{}\mathrm{\left(}{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{,}{H}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$.

Suppose that ${f}_{0}:\mathrm{\Omega }×ℝ\to ℝ$ is a Caratheodory function such that

with ${a}_{0}\in {L}^{\mathrm{\infty }}{\left(\mathrm{\Omega }\right)}_{+}$ and

(the critical Sobolev exponent for p). We set ${F}_{0}\left(z,x\right)={\int }_{0}^{x}{f}_{0}\left(z,s\right)𝑑s$ and we consider the ${C}^{1}$-functional ${\phi }_{0}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

The next result can be found in [18, p. 409].

#### Proposition 2.3.

If ${u}_{\mathrm{0}}\mathrm{\in }{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ is a local ${C}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{\right)}$-minimizer of ${\phi }_{\mathrm{0}}$, that is, there exists ${\rho }_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that

then ${u}_{\mathrm{0}}\mathrm{\in }{C}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{\right)}$ for some $\alpha \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ and ${u}_{\mathrm{0}}$ is also a local ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$-minimizer of ${\phi }_{\mathrm{0}}$, that is, there exists ${\rho }_{\mathrm{1}}\mathrm{>}\mathrm{0}$ such that

To make an effective use of Proposition 2.3, we need a strong comparison principle, which is provided by the next proposition and is a particular case of [9, Proposition 3].

Given ${h}_{1},{h}_{2}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ we write ${h}_{1}\prec {h}_{2}$ if for every $K\subseteq \mathrm{\Omega }$ compact, we can find $\epsilon =\epsilon \left(K\right)>0$ such that ${h}_{1}\left(z\right)+\epsilon \le {h}_{2}\left(z\right)$ for a.a. $z\in K$. Evidently, if ${h}_{1},{h}_{2}\in C\left(\mathrm{\Omega }\right)$, and ${h}_{1}\left(z\right)<{h}_{2}\left(z\right)$ for all $z\in \mathrm{\Omega }$, then ${h}_{1}\prec {h}_{2}$.

#### Proposition 2.4.

If $\xi \mathrm{,}{h}_{\mathrm{1}}\mathrm{,}{h}_{\mathrm{2}}\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{,}\xi \mathit{}\mathrm{\left(}z\mathrm{\right)}\mathrm{\ge }\mathrm{0}$ for a.a. $z\mathrm{\in }\mathrm{\Omega }\mathrm{,}{h}_{\mathrm{1}}\mathrm{\prec }{h}_{\mathrm{2}}\mathrm{,}u\mathrm{\in }{C}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{\right)}\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$, $v\mathrm{\in }\mathrm{int}\mathit{}{C}_{\mathrm{+}}$ and

$-{\mathrm{\Delta }}_{p}u\left(z\right)-\mathrm{\Delta }u\left(z\right)+\xi \left(z\right){|u\left(z\right)|}^{p-2}u\left(z\right)={h}_{1}\left(z\right)$$-{\mathrm{\Delta }}_{p}v\left(z\right)-\mathrm{\Delta }v\left(z\right)+\xi \left(z\right)v{\left(z\right)}^{p-1}={h}_{2}\left(z\right)$

then $v\mathrm{-}u\mathrm{\in }\mathrm{int}\mathit{}{C}_{\mathrm{+}}$.

Next we recall some basic facts about the spectrum of the Dirichlet p-Laplacian. So, we consider the following nonlinear eigenvalue problem:

(2.1)

We say that $\stackrel{^}{\lambda }\in ℝ$ is an eigenvalue of $\left(-{\mathrm{\Delta }}_{p},{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$ if problem (2.1) admits a nontrivial solution $\stackrel{^}{u}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, known as an eigenfunction corresponding to the eigenvalue $\stackrel{^}{\lambda }$. By $\stackrel{^}{\sigma }\left(p\right)\subseteq ℝ$ we denote the spectrum of $\left(-{\mathrm{\Delta }}_{p},{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$ (that is, the set of eigenvalues). We know that there exists a smallest eigenvalue ${\stackrel{^}{\lambda }}_{1}\left(p\right)\in ℝ$, which has the following properties:

• ${\stackrel{^}{\lambda }}_{1}\left(p\right)>0$ and it is isolated in the spectrum $\stackrel{^}{\sigma }\left(p\right)$ (that is, there exists $\epsilon >0$ such that $\left({\stackrel{^}{\lambda }}_{1}\left(p\right),{\stackrel{^}{\lambda }}_{1}\left(p\right)+\epsilon \right)\cap \stackrel{^}{\sigma }\left(p\right)=\mathrm{\varnothing }\right)$,

• ${\stackrel{^}{\lambda }}_{1}\left(p\right)$ is simple (that is, if ${\stackrel{^}{u}}_{1},{\stackrel{^}{u}}_{2}$ are eigenfunctions corresponding to ${\stackrel{^}{\lambda }}_{1}\left(p\right)$, then ${\stackrel{^}{u}}_{1}=\xi {\stackrel{^}{u}}_{2}$ with $\xi \ne 0$),

• we have

${\stackrel{^}{\lambda }}_{1}\left(p\right)=inf\left[\frac{{\parallel Du\parallel }_{p}^{p}}{{\parallel u\parallel }_{p}^{p}}:u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right),u\ne 0\right].$(2.2)

The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace. Evidently, the elements of this eigenspace have constant sign. By ${\stackrel{^}{u}}_{1}\left(p\right)$ we denote the positive ${L}^{p}$-normalized (that is, ${\parallel {\stackrel{^}{u}}_{1}\left(p\right)\parallel }_{p}=1$) eigenfunction corresponding to the eigenvalue ${\stackrel{^}{\lambda }}_{1}\left(p\right)>0$. The nonlinear regularity theory and the nonlinear maximum principle (see [10, pp. 737–738]) imply that ${\stackrel{^}{u}}_{1}\in \mathrm{int}{C}_{+}$. It is easy to see that $\stackrel{^}{\sigma }\left(p\right)\subseteq \left(0,+\mathrm{\infty }\right)$ is closed and using the Ljusternik–Schnirelmann minimax scheme, we produce a whole sequence of distinct eigenvalues ${\left\{{\stackrel{^}{\lambda }}_{k}\left(p\right)\right\}}_{k\in ℕ}$ of $\left(-{\mathrm{\Delta }}_{p},{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$ such that ${\stackrel{^}{\lambda }}_{k}\left(p\right)\to +\mathrm{\infty }$ as $k\to +\mathrm{\infty }$. These eigenvalues are known as “variational eigenvalues” and we do not know if they exhaust $\stackrel{^}{\sigma }\left(p\right)$. This is the case if $N=1$ (scalar eigenvalue problem) or if $p=2$ (linear eigenvalue problem). All eigenvalues $\stackrel{^}{\lambda }\ne {\stackrel{^}{\lambda }}_{1}\left(p\right)$ have nodal eigenfunctions.

In what follows, for notational simplicity, we write

${\stackrel{^}{\lambda }}_{1}={\stackrel{^}{\lambda }}_{1}\left(p\right)\mathit{ }\text{and}\mathit{ }{\stackrel{^}{u}}_{1}={\stackrel{^}{u}}_{1}\left(p\right)\in \mathrm{int}{C}_{+}.$

We will also encounter a weighted version of the eigenvalue problem (2.1). Namely, let $\eta \in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, with $\eta \left(z\right)\ge 0$ for a.a. $z\in \mathrm{\Omega }$, $\eta \not\equiv 0$. We consider the following nonlinear eigenvalue problem:

The same results are true for this problem. So, there exists a smallest eigenvalue ${\stackrel{~}{\lambda }}_{1}\left(p,\eta \right)>0$, which is isolated, simple and admits the following variational characterization:

${\stackrel{~}{\lambda }}_{1}\left(p,\eta \right)=inf\left[\frac{{\parallel Du\parallel }_{p}^{p}}{{\int }_{\mathrm{\Omega }}\eta \left(z\right){|u|}^{p}𝑑z}:u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right),u\ne 0\right].$

These properties lead to the following monotonicity property for the map $\eta ↦{\stackrel{~}{\lambda }}_{1}\left(p,\eta \right)$ (see [18, p. 250]).

#### Proposition 2.5.

If $\eta \mathrm{,}\stackrel{\mathrm{^}}{\eta }\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$, $\mathrm{0}\mathrm{\le }\eta \mathit{}\mathrm{\left(}z\mathrm{\right)}\mathrm{\le }\stackrel{\mathrm{^}}{\eta }\mathit{}\mathrm{\left(}z\mathrm{\right)}$ for a.a. $z\mathrm{\in }\mathrm{\Omega }\mathrm{,}\eta \mathrm{\not\equiv }\mathrm{0}\mathrm{,}\stackrel{\mathrm{^}}{\eta }\mathrm{\not\equiv }\eta$, then ${\stackrel{\mathrm{~}}{\lambda }}_{\mathrm{1}}\mathit{}\mathrm{\left(}p\mathrm{,}\stackrel{\mathrm{^}}{\eta }\mathrm{\right)}\mathrm{<}{\stackrel{\mathrm{~}}{\lambda }}_{\mathrm{1}}\mathit{}\mathrm{\left(}p\mathrm{,}\eta \mathrm{\right)}$.

Next let us recall some basic definitions and facts from the theory of critical groups (Morse theory). So, let X be a Banach space, $\phi \in {C}^{1}\left(X,ℝ\right)$ and $c\in ℝ$. We introduce the following sets:

${K}_{\phi }=\left\{u\in X:{\phi }^{\prime }\left(u\right)=0\right\}$${K}_{\phi }^{c}=\left\{u\in {K}_{\phi }:\phi \left(u\right)=c\right\}$${\phi }^{c}=\left\{u\in X:\phi \left(u\right)\le c\right\}$

Let $\left({Y}_{1},{Y}_{2}\right)$ be a topological pair such that ${Y}_{2}\subseteq {Y}_{1}\subseteq X$. For any $k\in {ℕ}_{0}$, by ${H}_{k}\left({Y}_{1},{Y}_{2}\right)$ we denote the kth-relative singular homology group with integer coefficients for the pair $\left({Y}_{1},{Y}_{2}\right)$. Recall that ${H}_{k}\left({Y}_{1},{Y}_{2}\right)=0$ for all $k\in -ℕ$. If $u\in {K}_{\phi }^{c}$ is isolated, then the critical groups of φ at u are defined by

with U being a neighborhood of u such that ${K}_{\phi }\cap {\phi }^{c}\cap U=\left\{u\right\}$. The excision property of singular homology implies that this definition of critical groups above is independent of the choice of the neighborhood U.

If $u\in {K}_{\phi }$ is isolated and of mountain pass-type (see Theorem 2.1), then ${C}_{1}\left(\phi ,u\right)\ne 0$. Moreover, if $\phi \in {C}^{2}\left(X,ℝ\right)$, then from [21] we know that

with ${\delta }_{k,m}$ being the Kronecker symbol defined by

Next we fix our notation. Given $x\in ℝ$, we set ${x}^{±}=\mathrm{max}\left\{±x,0\right\}$. Then for $u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, we define

${u}^{±}\left(\cdot \right)=u{\left(\cdot \right)}^{±}.$

We know that ${u}^{±}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, $u={u}^{+}-{u}^{-}$, $|u|={u}^{+}+{u}^{-}$. Also, given a measurable function $g:\mathrm{\Omega }×ℝ\to ℝ$ (for example, a Caratheodory function), we define

which is the Nemytskii operator corresponding to g. By $|\cdot {|}_{N}$ we denote the Lebesgue measure on ${ℝ}^{N}$, and if $1, then $\frac{1}{r}+\frac{1}{{r}^{\prime }}=1$.

If $u,\stackrel{^}{u}\in {W}^{1,p}\left(\mathrm{\Omega }\right)$ and $u\le \stackrel{^}{u}$, then we define

By ${\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[u,\stackrel{^}{u}\right]$, we denote the interior of $\left[u,\stackrel{^}{u}\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ in the ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$-norm. Also, if $k:\left(0,+\mathrm{\infty }\right)\to {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$, then we say that $k\left(\cdot \right)$ is strictly increasing if $\lambda <{\lambda }^{\prime }$ implies that $k\left({\lambda }^{\prime }\right)-k\left(\lambda \right)\in \mathrm{int}{C}_{+}$. Finally, if $u\in {W}^{1,p}\left(\mathrm{\Omega }\right)$, then

Now we can introduce the hypotheses on the perturbation term $f\left(z,x\right)$.

#### Hypothesis 2.6.

$f:\mathrm{\Omega }×ℝ\to ℝ$ is a Caratheodory function such that $f\left(z,0\right)=0$ for a.a. $z\in \mathrm{\Omega }$, which satisfies the following properties:

• (i)

For every $\rho >0$, there exists ${a}_{\rho }\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ such that

• (ii)

We have

• (iii)

If $F\left(z,x\right)={\int }_{0}^{x}f\left(z,s\right)𝑑s$, then

• (iv)

There exist functions ${w}_{±}\in {W}^{1,p}\left(\mathrm{\Omega }\right)\cap {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, with ${\mathrm{\Delta }}_{p}{w}_{±},\mathrm{\Delta }{w}_{±}\in {L}^{{p}^{\prime }}\left(\mathrm{\Omega }\right)$, and ${\theta }_{±}>0$ such that, for a.a. $z\in \mathrm{\Omega }$,

${w}_{-}\left(z\right)\le -{c}_{-}<0<{c}_{+}\le {w}_{+}\left(z\right),$$-{\mathrm{\Delta }}_{p}{w}_{-}\left(z\right)-\mathrm{\Delta }{w}_{-}\left(z\right)\le 0\le -{\mathrm{\Delta }}_{p}{w}_{+}\left(z\right)-\mathrm{\Delta }{w}_{+}\left(z\right),$${\theta }_{+}{w}_{+}{\left(z\right)}^{q-1}+f\left(z,{w}_{+}\left(z\right)\right)\le -{\stackrel{^}{c}}_{+}<0\le {\stackrel{^}{c}}_{-}\le {\theta }_{-}{|{w}_{-}\left(z\right)|}^{q-2}{w}_{-}\left(z\right)+f\left(z,{w}_{-}\left(z\right)\right).$

• (v)

If ${c}_{0}=\mathrm{min}\left\{{c}_{+},{c}_{-}\right\}>0$, then we can find ${\delta }_{0}\in \left(0,{c}_{0}\right)$ such that for all $K\subseteq \mathrm{\Omega }$ compact and all $0, we have

• (vi)

For every $\rho >0$, there exists ${\stackrel{^}{\xi }}_{\rho }>0$ such that for a.a. $z\in \mathrm{\Omega }$, the function

$x↦f\left(z,x\right)+{\stackrel{^}{\xi }}_{\rho }{|x|}^{p-2}x$

is nondecreasing on $\left[-\rho ,\rho \right]$.

#### Remark 2.7.

Hypotheses 2.6 (i), (ii) imply that

Hypothesis 2.6 (ii) says that at $±\mathrm{\infty }$ we can have resonance with respect to the principal eigenvalue ${\stackrel{^}{\lambda }}_{1}>0$. In the process of the proof we shall see that Hypothesis 2.6 (iii) implies that the resonance occurs from the right of ${\stackrel{^}{\lambda }}_{1}>0$ in the sense that

This makes the problem noncoercive and so the direct method of the calculus of variations is not directly applicable on (1.1). Hypothesis 2.6 (iv) is satisfied if we can find ${t}_{-}<0<{t}_{+}$ such that

Therefore, Hypotheses 2.6 (iv), (v) dictate an oscillatory behavior for $f\left(z,\cdot \right)$ near zero. Hypothesis 2.6 (vi) is satisfied if, for example, for a.a. $z\in \mathrm{\Omega }$, $f\left(z,\cdot \right)$ is differentiable, and for every $\rho >0$, we can find ${\stackrel{^}{\xi }}_{\rho }>0$ such that

#### Example 2.8.

The following function satisfies Hypothesis 2.6 (for the sake of simplicity, we drop the z-dependence):

with $1<\tau <\eta <\mathrm{\infty }$, $\beta \ge {\stackrel{^}{\lambda }}_{1}$ and $1<\gamma \le 2<\mu .

## 3 Solutions of constant sign

In this section we produce solutions of constant sign (positive and negative solutions) of problem (1.1).

We start by considering the following auxiliary nonlinear parametric Dirichlet $\left(p,2\right)$-equation:

(3.1)

#### Proposition 3.1.

For every $\lambda \mathrm{>}\mathrm{0}$, problem (3.1) has a unique positive solution ${\stackrel{\mathrm{~}}{u}}_{\lambda }\mathrm{\in }\mathrm{int}\mathit{}{C}_{\mathrm{+}}$, and since (3.1) is odd, ${\stackrel{\mathrm{~}}{v}}_{\lambda }\mathrm{=}\mathrm{-}{\stackrel{\mathrm{~}}{u}}_{\lambda }\mathrm{\in }\mathrm{-}\mathrm{int}\mathit{}{C}_{\mathrm{+}}$ is the unique negative solution of (3.1); moreover, $\lambda \mathrm{↦}{\stackrel{\mathrm{~}}{u}}_{\lambda }$ is strictly increasing and

#### Proof.

The existence of a positive solution for problem (3.1) can be established using the direct method of the calculus of variations. More precisely, let ${\psi }_{\lambda }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ be the ${C}^{1}$-functional defined by

Since $1, we see that ${\psi }_{\lambda }\left(\cdot \right)$ is coercive. Also, using the Sobolev embedding theorem, we show that ${\psi }_{\lambda }\left(\cdot \right)$ is sequentially weakly lower semicontinuous. So, using the Weierstrass–Tonelli theorem, we can find ${\stackrel{~}{u}}_{\lambda }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

${\psi }_{\lambda }\left({\stackrel{~}{u}}_{\lambda }\right)=inf\left[{\psi }_{\lambda }\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right].$(3.2)

Since $q<2, for $t\in \left(0,1\right)$ small, we have ${\psi }_{\lambda }\left(t{\stackrel{^}{u}}_{1}\right)<0$, hence ${\psi }_{\lambda }\left({u}_{\lambda }\right)<0={\psi }_{\lambda }\left(0\right)$ (see (3.2)), and thus ${u}_{\lambda }\ne 0$. From (3.2) we have ${\psi }_{\lambda }^{\prime }\left({\stackrel{~}{u}}_{\lambda }\right)=0,$ that is,

(3.3)

In (3.3) we choose $h=-{\stackrel{~}{u}}_{\lambda }^{-}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then ${\parallel D{\stackrel{~}{u}}_{\lambda }^{-}\parallel }_{p}^{p}+{\parallel D{\stackrel{~}{u}}_{\lambda }^{-}\parallel }_{2}^{2}=0$, and thus ${\stackrel{~}{u}}_{\lambda }\ge 0$ and ${\stackrel{~}{u}}_{\lambda }\ne 0$. From (3.3) it follows that

therefore ${\stackrel{~}{u}}_{\lambda }$ is a positive solution of (3.1).

From [18, Corollary 8.6, p. 208], we have ${\stackrel{~}{u}}_{\lambda }\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right).$ Then, invoking [15, Theorem 1], we infer that ${\stackrel{~}{u}}_{\lambda }\in {C}_{+}\setminus \left\{0\right\}.$ The nonlinear strong maximum principle of Pucci and Serrin [26, pp. 111, 120] implies that ${\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$.

Next we show that this positive solution is unique. To this end, we introduce the integral functional $j:{L}^{1}\left(\mathrm{\Omega }\right)\to \overline{ℝ}=ℝ\cup \left\{+\mathrm{\infty }\right\}$ defined by

From [7, Lemma 1], we know that $j\left(\cdot \right)$ is convex.

Suppose that ${\stackrel{~}{y}}_{\lambda }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is another positive solution of (3.1). Again we have ${\stackrel{~}{y}}_{\lambda }\in \mathrm{int}{C}_{+}$. Then for any $h\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ and for $|t|<1$ small, we have

${\stackrel{~}{u}}_{\lambda }+th\in \mathrm{dom}j\mathit{ }\text{and}\mathit{ }{\stackrel{~}{y}}_{\lambda }+th\in \mathrm{dom}j,$

where $\mathrm{dom}j=\left\{u\in {L}^{1}\left(\mathrm{\Omega }\right):j\left(u\right)<+\mathrm{\infty }\right\}$ (the effective domain of $j\left(\cdot \right)$). We can easily see that $j\left(\cdot \right)$ is Gateaux differentiable at ${\stackrel{~}{u}}_{\lambda }^{2}$ and at ${\stackrel{~}{y}}_{\lambda }^{2}$ in the direction of h. Moreover, the chain rule and Green’s identity (see [10, p. 211]) imply that

The convexity of $j\left(\cdot \right)$ implies the monotonicity of ${j}^{\prime }\left(\cdot \right)$. Therefore,

$0\le {\int }_{\mathrm{\Omega }}\left(\frac{-{\mathrm{\Delta }}_{p}{\stackrel{~}{u}}_{\lambda }-\mathrm{\Delta }{\stackrel{~}{u}}_{\lambda }}{{\stackrel{~}{u}}_{\lambda }}-\frac{{\mathrm{\Delta }}_{p}{\stackrel{~}{y}}_{\lambda }-\mathrm{\Delta }{\stackrel{~}{y}}_{\lambda }}{{\stackrel{~}{y}}_{\lambda }}\right)\left({\stackrel{~}{u}}_{\lambda }-{\stackrel{~}{y}}_{\lambda }\right)𝑑z={\int }_{\mathrm{\Omega }}\lambda \left[\frac{1}{{\stackrel{~}{u}}_{\lambda }^{2-q}}-\frac{1}{{\stackrel{~}{y}}_{\lambda }^{2-q}}\right]\left({\stackrel{~}{u}}_{\lambda }-{\stackrel{~}{y}}_{\lambda }\right)𝑑z,$

and hence ${\stackrel{~}{u}}_{\lambda }={\stackrel{~}{y}}_{\lambda }$ (since $q<2$). This proves the uniqueness of the positive solution of problem (3.1).

Let $0<\eta <\lambda$ and consider the Caratheodory function

(3.4)

We set ${K}_{\eta }\left(z,x\right)={\int }_{0}^{x}{k}_{\eta }\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\stackrel{^}{\psi }}_{\eta }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

Evidently ${\stackrel{^}{\psi }}_{\eta }\left(\cdot \right)$ is coercive (see (3.4)) and sequentially weakly lower semicontinuous. So, we can find ${\overline{u}}_{\eta }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

${\stackrel{^}{\psi }}_{\eta }\left({\overline{u}}_{\eta }\right)=inf\left[{\stackrel{^}{\psi }}_{\eta }\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right].$(3.5)

Since $1, we see that ${\stackrel{^}{\psi }}_{\eta }\left({\overline{u}}_{\eta }\right)<0={\stackrel{^}{\psi }}_{\eta }\left(0\right)$, therefore ${\overline{u}}_{\eta }\ne 0$.

From (3.5) we have ${\stackrel{^}{\psi }}_{\eta }^{\prime }\left({\overline{u}}_{\eta }\right)=0,$ that is,

(3.6)

In (3.6), first we choose $h=-{\overline{u}}_{\eta }^{-}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then ${\parallel D{\overline{u}}_{\eta }^{-}\parallel }_{p}^{p}+{\parallel D{\overline{u}}_{\eta }^{-}\parallel }_{2}^{2}=0$ (see (3.4)), hence ${\overline{u}}_{\eta }\ge 0$, ${\overline{u}}_{\eta }\ne 0$. Also, in (3.6), we choose $h={\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (from (3.4) and since $\eta <\lambda$)

$〈{A}_{p}\left({\overline{u}}_{\eta }\right),{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}〉+〈A\left({\overline{u}}_{\eta }\right),{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}〉={\int }_{\mathrm{\Omega }}\eta {\stackrel{~}{u}}_{\lambda }^{q-1}{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}𝑑z$$\le {\int }_{\mathrm{\Omega }}\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}𝑑z$$=〈{A}_{p}\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}〉+〈A\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\overline{u}}_{\eta }-{\stackrel{~}{u}}_{\lambda }\right)}^{+}〉,$

hence ${\overline{u}}_{\eta }\le {\stackrel{~}{u}}_{\lambda }$ (see Proposition 2.2).

So, we have proved that

${\overline{u}}_{\eta }\in \left[0,{\stackrel{~}{u}}_{\lambda }\right],{\overline{u}}_{\eta }\ne 0.$(3.7)

From (3.4), (3.6) and (3.7), we infer that ${\overline{u}}_{\eta }={\stackrel{~}{u}}_{\eta }\in \mathrm{int}{C}_{+}$, thus

${\stackrel{~}{u}}_{\eta }\le {\stackrel{~}{u}}_{\lambda }.$(3.8)

We have (see (3.8) and recall that $\eta <\lambda$, ${\stackrel{~}{u}}_{\eta }\in \mathrm{int}{C}_{+}$)

(3.9)

Let ${h}_{1}=\eta {\stackrel{~}{u}}_{\eta }^{q-1}$ and ${h}_{2}=\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}$. Evidently ${h}_{1},{h}_{2}\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$. For all $K\subseteq \mathrm{\Omega }$ compact, let ${c}_{K}={\mathrm{min}}_{K}{\stackrel{~}{u}}_{\eta }>0$ (recall ${\stackrel{~}{u}}_{\eta }\in \mathrm{int}{C}_{+}$). Then (see (3.8))

From (3.9) and Proposition 2.4 it follows that ${\stackrel{~}{u}}_{\lambda }-{\stackrel{~}{u}}_{\eta }\in \mathrm{int}{C}_{+}$, therefore $\lambda ↦{\stackrel{~}{u}}_{\lambda }$ is strictly increasing from $\left(0,+\mathrm{\infty }\right)$ into ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$.

Finally, let $\lambda >0$ and let ${\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ be the unique solution of (3.1). Then we have

Choosing $h={\stackrel{~}{u}}_{\lambda }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, we obtain ${\parallel D{\stackrel{~}{u}}_{\lambda }\parallel }_{p}^{p}\le \lambda {\parallel {\stackrel{~}{u}}_{\lambda }\parallel }_{q}^{q}$, hence ${\parallel {\stackrel{~}{u}}_{\lambda }\parallel }^{p-q}\le \lambda {c}_{2}$ for some ${c}_{2}>0$ (recall that $q). Therefore, given $\mu >0$, we see that

(3.10)

Invoking [18, Corollary 8.6, p. 208], we can find ${c}_{3}>0$ such that

Then [15, Theorem 1] implies that there exist $\alpha \in \left(0,1\right)$ and ${c}_{4}>0$ such that

(3.11)

From (3.10), (3.11) and the compact embedding of ${C}_{0}^{1,\alpha }\left(\overline{\mathrm{\Omega }}\right)$ into ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$, we conclude that

Since problem (3.1) is odd, ${\stackrel{~}{v}}_{\lambda }=-{\stackrel{~}{u}}_{\lambda }\in -\mathrm{int}{C}_{+}$ is the unique negative solution of (3.1) for all $\lambda >0$. Also, $\lambda ↦{\stackrel{~}{v}}_{\lambda }$ is strictly decreasing from $\left(0,+\mathrm{\infty }\right)$ into ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ and ${\parallel {\stackrel{~}{v}}_{\lambda }\parallel }_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\to 0$ as $\lambda \to {0}^{+}$. ∎

Let ${\delta }_{0}>0$ be as postulated by Hypothesis 2.6 (v). On account of Proposition 2.5, we can find ${\lambda }_{±}>0$ such that

${\stackrel{~}{u}}_{\lambda }\left(z\right)\in \left[0,{\delta }_{0}\right]$${\stackrel{~}{v}}_{\lambda }\left(z\right)\in \left[-{\delta }_{0},0\right]$(3.12)

With ${\theta }_{±}>0$ as in Hypothesis 2.6 (iv), we set

${\lambda }_{+}^{*}=\mathrm{min}\left\{{\lambda }_{+},{\theta }_{+}\right\}\mathit{ }\text{and}\mathit{ }{\lambda }_{-}^{*}=\mathrm{min}\left\{{\lambda }_{-},{\theta }_{-}\right\}.$

#### Proposition 3.2.

If Hypothesis 2.6 holds, then

• (a)

for all $0<\lambda \le {\lambda }_{+}^{*}$ , problem ( 1.1 ) admits a positive solution ${u}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right]$,

• (b)

for all $0<\lambda \le {\lambda }_{-}^{*}$ , problem ( 1.1 ) admits a negative solution ${v}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{w}_{-},{\stackrel{~}{v}}_{\lambda }\right].$

#### Proof.

(a) Recall that ${\delta }_{0}<{c}_{0}$ (see Hypothesis 2.6 (v)). This fact and (3.12) permit the definition of the Caratheodory function

(3.13)

We set ${E}_{\lambda }^{+}\left(z,x\right)={\int }_{0}^{x}{e}_{\lambda }^{+}\left(z,x\right)𝑑s$ and consider the ${C}^{1}$-functional ${\stackrel{^}{\phi }}_{\lambda }^{+}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

From (3.13) it is clear that ${\stackrel{^}{\phi }}_{\lambda }^{+}\left(\cdot \right)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ${u}_{0}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that ${\stackrel{^}{\phi }}_{\lambda }^{+}\left({u}_{0}\right)=inf\left[{\stackrel{^}{\phi }}_{\lambda }^{+}\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right]$, hence ${\left({\stackrel{^}{\phi }}_{\lambda }^{+}\right)}^{\prime }\left({u}_{0}\right)=0$, and therefore

(3.14)

In (3.14), first we choose $h={\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (see (3.12), (3.13), Hypothesis 2.6 (v) and Proposition 3.1)

$〈{A}_{p}\left({u}_{0}\right),{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}〉+〈A\left({u}_{0}\right),{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}〉={\int }_{\mathrm{\Omega }}\left[\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}+f\left(z,{\stackrel{~}{u}}_{\lambda }\right)\right]{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}𝑑z$$\ge {\int }_{\mathrm{\Omega }}\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}𝑑z$$=〈{A}_{p}\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}〉+〈A\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\stackrel{~}{u}}_{\lambda }-{u}_{0}\right)}^{+}〉,$

hence ${\stackrel{~}{u}}_{\lambda }\le {u}_{0}$ (see Proposition 2.2).

Next in (3.14) we choose $h={\left({u}_{0}-{w}_{+}\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (see (3.13), Hypothesis 2.6 (iv) and recall that $0<\lambda \le {\lambda }_{+}^{*}\le {\theta }_{+}$)

$〈{A}_{p}\left({u}_{0}\right),{\left({u}_{0}-{w}_{+}\right)}^{+}〉+〈A\left({u}_{0}\right),{\left({u}_{0}-{w}_{+}\right)}^{+}〉={\int }_{\mathrm{\Omega }}\left[\lambda {w}_{+}^{q-1}+f\left(z,{w}_{+}\right)\right]{\left({u}_{0}-{w}_{+}\right)}^{+}𝑑z$$\le {\int }_{\mathrm{\Omega }}\left[{\theta }_{+}{w}_{+}^{q-1}+f\left(z,{w}_{+}\right)\right]{\left({u}_{0}-{w}_{+}\right)}^{+}𝑑x$$\le 〈{A}_{p}\left({w}_{+}\right),{\left({u}_{0}-{w}_{+}\right)}^{+}〉+〈A\left({w}_{+}\right),{\left({u}_{0}-{w}_{+}\right)}^{+}〉,$

thus ${u}_{0}\le {w}_{+}$ (see Proposition 2.2).

We have proved that

${u}_{0}\in \left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right].$(3.15)

From (3.13), (3.14), (3.15), it follows that

(3.16)

From (3.15), (3.16) and [15, Theorem 1], we infer that

${u}_{0}\in \left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right]\cap \mathrm{int}{C}_{+}.$(3.17)

Now let $\rho ={\parallel {u}_{0}\parallel }_{\mathrm{\infty }}$ and let ${\stackrel{^}{\xi }}_{\rho }>0$ be as postulated by Hypothesis 2.6 (vi). Then we have (see (3.12), (3.16), (3.17), Hypotheses 2.6 (v)–(vi) and Proposition 3.1)

$-{\mathrm{\Delta }}_{p}{u}_{0}\left(z\right)-\mathrm{\Delta }{u}_{0}\left(z\right)+{\stackrel{^}{\xi }}_{\rho }{u}_{0}{\left(z\right)}^{p-1}=\lambda {u}_{0}{\left(z\right)}^{q-1}+f\left(z,{u}_{0}\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{u}_{0}{\left(z\right)}^{p-1}$$\ge \lambda {\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{q-1}+f\left(z,{\stackrel{~}{u}}_{\lambda }\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{p-1}$$\ge \lambda {\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{q-1}+{\stackrel{^}{\xi }}_{\rho }{\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{p-1}$(3.18)

Set

${h}_{1}\left(z\right)=\lambda {\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{q-1}+{\stackrel{^}{\xi }}_{\rho }{\stackrel{~}{u}}_{\lambda }{\left(z\right)}^{p-1},{h}_{2}\left(z\right)=\lambda {u}_{0}{\left(z\right)}^{q-1}+f\left(z,{u}_{0}\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{u}_{0}{\left(z\right)}^{p-1}.$

Evidently ${h}_{1},{h}_{2}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. Also, for $K\subseteq \mathrm{\Omega }$ compact, we have (recall that ${u}_{0}\in \mathrm{int}{C}_{+}$)

Therefore (see (3.17) and Hypothesis 2.6 (v)),

Then (3.18) and Proposition 2.4 imply that ${u}_{0}-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}.$

Also, we have (see (3.16), (3.17), Hypotheses 2.6 (iv), (vi), and recall that $0<\lambda \le {\theta }_{+}$)

$-{\mathrm{\Delta }}_{p}{u}_{0}\left(z\right)-\mathrm{\Delta }{u}_{0}\left(z\right)+{\stackrel{^}{\xi }}_{\rho }{u}_{0}{\left(z\right)}^{p-1}=\lambda {u}_{0}{\left(z\right)}^{q-1}+f\left(z,{u}_{0}\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{u}_{0}{\left(z\right)}^{p-1}$$\le {\theta }_{+}{w}_{+}{\left(z\right)}^{q-1}+f\left(z,{w}_{+}\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{w}_{+}{\left(z\right)}^{p-1}$$\le -{\stackrel{^}{c}}_{1}+{\stackrel{^}{\xi }}_{\rho }{w}_{+}{\left(z\right)}^{p-1}$(3.19)

From (3.19) and Proposition 2.4, we have ${w}_{+}-{u}_{0}\in \mathrm{int}{C}_{+}.$ We conclude that ${u}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right].$

(b) In the negative semiaxis, we consider the Caratheodory function

(3.20)

We set ${E}_{\lambda }^{-}\left(z,x\right)={\int }_{0}^{x}{e}_{\lambda }^{-}\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\stackrel{^}{\phi }}_{\lambda }^{-}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

Arguing as in part (a), using this time the functional ${\stackrel{^}{\phi }}_{\lambda }^{-}$ and (3.20), we produce a negative solution ${v}_{0}$ for problem (1.1), with $\lambda \in \left(0,{\lambda }_{-}^{*}\right]$, such that ${v}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{w}_{-},{\stackrel{~}{v}}_{\lambda }\right].$

Next, using ${u}_{0}\in \mathrm{int}{C}_{+}$, ${v}_{0}\in -\mathrm{int}{C}_{+}$ together with suitable variational, truncation and comparison arguments, we will generate a second pair of constant sign smooth solutions.

#### Proposition 3.3.

If Hypothesis 2.6 holds, then

• (a)

for all $0<\lambda \le {\lambda }_{+}^{*}$ , problem ( 1.1 ) has a second positive solution $\stackrel{^}{u}\in \mathrm{int}{C}_{+}$ , with $\stackrel{^}{u}\ne {u}_{0}$, $\stackrel{^}{u}-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ ;

• (b)

for all $0<\lambda \le {\lambda }_{-}^{*}$ , problem ( 1.1 ) has a second negative solution $\stackrel{^}{v}\in -\mathrm{int}{C}_{+}$ , with $\stackrel{^}{v}\ne {v}_{0}$, ${\stackrel{~}{v}}_{\lambda }-\stackrel{^}{v}\in \mathrm{int}{C}_{+}$.

#### Proof.

(a) Let ${\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ be the unique positive solution of (3.1) (see Proposition 3.1). We introduce the following truncation of the reaction in problem (1.1):

(3.21)

This is a Caratheodory function. Set ${R}_{\lambda }^{+}\left(z,x\right)={\int }_{0}^{x}{r}_{\lambda }^{+}\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\phi }_{\lambda }^{+}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

From (3.13) and (3.21), it is clear that

${{\phi }_{\lambda }^{+}|}_{\left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right]}={{\stackrel{^}{\phi }}_{\lambda }^{+}|}_{\left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right]}.$(3.22)

Let ${u}_{0}\in \mathrm{int}{C}_{+}$ be the positive solution of problem (1.1) produced in Proposition 3.2. From the proof of that proposition, we know that ${u}_{0}\in \mathrm{int}{C}_{+}$ is a minimizer of ${\stackrel{^}{\phi }}_{\lambda }^{+}$, and ${u}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{\stackrel{~}{u}}_{\lambda },{w}_{+}\right]$. Hence, from (3.22), it follows that ${u}_{0}\in \mathrm{int}{C}_{+}$ is a local ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$-minimizer of ${\phi }_{\lambda }^{+}$, and thus (see Proposition 2.3)

(3.23)

Claim 1: ${K}_{{\phi }_{\lambda }^{+}}\subseteq \left[{\stackrel{~}{u}}_{\lambda }\right)\cap \mathrm{int}{C}_{+}$ and $u-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ for all $u\in {K}_{{\phi }_{\lambda }^{+}}$.

Let $u\in {K}_{{\phi }_{\lambda }^{+}}$. Then

(3.24)

In (3.24), we choose $h={\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (see (3.21), (3.12), Hypothesis 2.6 (v) and Proposition 3.1)

$〈{A}_{p}\left(u\right),{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}〉+〈A\left(u\right),{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}〉={\int }_{\mathrm{\Omega }}\left[\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}+f\left(z,{\stackrel{~}{u}}_{\lambda }\right)\right]{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}𝑑z$$\ge {\int }_{\mathrm{\Omega }}\lambda {\stackrel{~}{u}}_{\lambda }^{q-1}{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}𝑑z$$=〈{A}_{p}\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}〉+〈A\left({\stackrel{~}{u}}_{\lambda }\right),{\left({\stackrel{~}{u}}_{\lambda }-u\right)}^{+}〉,$

thus ${\stackrel{~}{u}}_{\lambda }\le u$ (see Proposition 2.2).

The nonlinear regularity theory of Lieberman [15] implies $u\in \mathrm{int}{C}_{+}$. In addition, as in the proof of Proposition 3.2, using Proposition 2.4, we show that $u-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$. This proves claim 1.

On account of claim 1, we may assume that

(3.25)

Otherwise, we already have an infinity of smooth solutions of problem (1.1) satisfying $u-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ for all the solutions u (see (3.21) and claim 1). Hence, we are done.

From (3.23) and (3.25) it follows that we can find $\rho \in \left(0,1\right)$ small such that

${\phi }_{\lambda }^{+}\left({u}_{0}\right)(3.26)

(see the proof of [1, Proposition 29]).

From Hypothesis 2.6 (iii), we see that given $\beta >0$, we can find ${M}_{1}={M}_{1}\left(\beta \right)>0$ such that

Then we have

hence

(3.27)

From Hypothesis 2.6 (ii), we have

(3.28)

So, if in (3.27) we let $x\to ±\mathrm{\infty }$ and use (3.28), then

$\frac{{\stackrel{^}{\lambda }}_{1}}{p}-\frac{F\left(z,y\right)}{{|y|}^{p}}\le -\frac{\beta }{p-2}\frac{1}{{|y|}^{p-2}},$

and thus

Since $\beta >0$ is arbitrary, we conclude that

(3.29)

Claim 2: ${\phi }_{\lambda }^{+}\left(t{\stackrel{^}{u}}_{1}\right)\to -\mathrm{\infty }$ as $t\to +\mathrm{\infty }$.

For $t>0$, we have

$p{\phi }_{\lambda }^{+}\left(t{\stackrel{^}{u}}_{1}\right)={t}^{p}{\stackrel{^}{\lambda }}_{1}+\frac{p{t}^{2}}{2}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{2}^{2}-{\int }_{\mathrm{\Omega }}{R}_{\lambda }^{+}\left(z,t{\stackrel{^}{u}}_{1}\right)𝑑z$$\le {t}^{p}{\stackrel{^}{\lambda }}_{1}+\frac{p{t}^{2}}{2}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{2}^{2}-\frac{p\lambda {t}^{q}}{q}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{q}^{q}-{\int }_{\mathrm{\Omega }}pF\left(z,t{\stackrel{^}{u}}_{1}\right)𝑑z+{c}_{4}$$={\int }_{\mathrm{\Omega }}\left[{\stackrel{^}{\lambda }}_{1}{\left(t{\stackrel{^}{u}}_{1}\right)}^{p}-pF\left(z,t{\stackrel{^}{u}}_{1}\right)\right]𝑑z+\frac{p{t}^{2}}{2}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{2}^{2}-\frac{p}{q}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{q}^{q}+{c}_{4}$

for some ${c}_{4}>0$ (see (3.21)), therefore

$\frac{p{\phi }_{\lambda }^{+}\left(t{\stackrel{^}{u}}_{1}\right)}{{t}^{2}}={\int }_{\mathrm{\Omega }}\frac{\left[{\stackrel{^}{\lambda }}_{1}{\left(t{\stackrel{^}{u}}_{1}\right)}^{p}-pF\left(z,t{\stackrel{^}{u}}_{1}\right)\right]}{{\left(t{\stackrel{^}{u}}_{1}\right)}^{2}}{\stackrel{^}{u}}_{1}^{2}𝑑z+\frac{p}{2}{\parallel D\left(t{\stackrel{^}{u}}_{1}\right)\parallel }_{2}^{2}-\frac{p}{q}\frac{1}{{\lambda }^{2-q}}{\parallel D{\stackrel{^}{u}}_{1}\parallel }_{q}^{q}+\frac{{c}_{4}}{{t}^{2}}.$

Passing to the limit as $t\to +\mathrm{\infty }$ and using (3.29), Fatou’s lemma and the fact that $q<2$, we obtain

hence

This proves claim 2.

Claim 3: ${\phi }_{\lambda }^{+}$ satisfies the C-condition.

We consider a sequence ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

(3.30)

and

(3.31)

From (3.31) we have

$|〈{A}_{p}\left({u}_{n}\right),h〉+〈A\left({u}_{n}\right),h〉-{\int }_{\mathrm{\Omega }}{r}_{\lambda }^{+}\left(z,{u}_{n}\right)h𝑑z|\le \frac{{\epsilon }_{n}\parallel h\parallel }{1+\parallel {u}_{n}\parallel }$(3.32)

for all $h\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, with ${\epsilon }_{n}\to {0}^{+}$.

In (3.32), we choose $h=-{v}_{n}^{-}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Using (3.21), we have

which implies ${\parallel {u}_{n}^{-}\parallel }^{p}\le {c}_{5}\parallel {u}_{n}^{-}\parallel$ for some ${c}_{5}>0$ and all $n\in ℕ$ (see Hypothesis 2.6 (i)), hence

(3.33)

Next we show that $\left\{{u}_{n}^{+}\right\}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded. Arguing by contradiction, we assume that at least for a subsequence, we have $\parallel {u}_{n}^{+}\parallel \to +\mathrm{\infty }$ as $n\to \mathrm{\infty }$. We let ${y}_{n}=\frac{{u}_{n}^{+}}{\parallel {u}_{n}^{+}\parallel },n\in ℕ$. Then $\parallel {y}_{n}\parallel =1,{y}_{n}\ge 0$ for all $n\in ℕ$. We may assume that

Multiplying (3.32) with $\frac{1}{{\parallel {u}_{n}^{+}\parallel }^{p-1}}$ and using (3.33), we obtain

$|〈{A}_{p}\left({y}_{n}\right),h〉+\frac{〈A\left({y}_{n}\right),h〉}{{\parallel {u}_{n}^{+}\parallel }^{p-2}}-{\int }_{\mathrm{\Omega }}\frac{{r}_{\lambda }^{+}\left(z,{u}_{n}^{+}\right)}{{\parallel {u}_{n}^{+}\parallel }^{p-1}}h𝑑z|\le {\epsilon }_{n}^{\prime }\parallel h\parallel$

for all $h\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, with ${\epsilon }_{n}^{\prime }\to {0}^{+}$. Using (3.21), we have

$|〈{A}_{p}\left({y}_{n}\right),h〉+\frac{〈A\left({y}_{n}\right),h〉}{{\parallel {u}_{n}^{+}\parallel }^{p-2}}-{\int }_{\mathrm{\Omega }}\frac{\left[\lambda {\left({u}_{n}^{+}\right)}^{q-1}+f\left(z,{u}_{n}^{+}\right)\right]}{{\parallel {u}_{n}^{+}\parallel }^{p-1}}h𝑑z|\le {\epsilon }_{n}^{\prime \prime }\parallel h\parallel$(3.34)

for all $h\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, with ${\epsilon }_{n}^{\prime \prime }\to {0}^{+}$. On account of (3.2), we see that

${\left\{\frac{\lambda {\left({u}_{n}^{+}\right)}^{q-1}+f\left(z,{u}_{n}^{+}\right)}{{\parallel {u}_{n}^{+}\parallel }^{p-1}}\right\}}_{n\ge 1}\subseteq {L}^{{p}^{\prime }}\left(\mathrm{\Omega }\right)$

is bounded. Therefore, we may assume that

(3.35)

with ${\stackrel{^}{\lambda }}_{1}\le \eta \left(z\right)\le \stackrel{^}{\eta }$ for a.a. $z\in \mathrm{\Omega }$ (see Hypothesis 2.6 (ii) and recall that $q<2).

In (3.34), we choose $h={y}_{n}-y\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Passing to the limit as $n\to \mathrm{\infty }$, we obtain (recall that $2)

$\underset{n\to \mathrm{\infty }}{lim}〈{A}_{p}\left({y}_{n}\right),{y}_{n}-y〉=0,$

therefore (see Proposition 2.2)

(3.36)

So, if in (3.34) we pass to the limit as $n\to \mathrm{\infty }$ and use (3.35), (3.36), then

and hence

(3.37)

Suppose that ${\eta }_{0}\not\equiv {\stackrel{^}{\lambda }}_{1}$ (see (3.35)). We have ${\stackrel{~}{\lambda }}_{1}\left(p,{\eta }_{0}\right)<{\stackrel{~}{\lambda }}_{1}\left(p,{\stackrel{^}{\lambda }}_{1}\right)=1$ (see Proposition 2.5), and thus y must be nodal (see (3.37)), which contradicts (3.36).

Next assume that ${\eta }_{0}\left(z\right)={\stackrel{^}{\lambda }}_{1}$ for a.a. $z\in \mathrm{\Omega }$. From (3.37) it follows that $y=\xi {\stackrel{^}{u}}_{1}\in \mathrm{int}{C}_{+}$ ($\xi >0$), hence $y\left(z\right)>0$ for all $z\in \mathrm{\Omega }$, and thus ${u}_{n}^{+}\left(z\right)\to +\mathrm{\infty }$ for all $z\in \mathrm{\Omega }$. Therefore (see Hypothesis 2.6 (iii)),

and thus (by Fatou’s lemma)

${\int }_{\mathrm{\Omega }}\frac{f\left(z,{u}_{n}^{+}\right){u}_{n}^{+}-pF\left(z,{u}_{n}^{+}\right)}{{\parallel {u}_{n}^{+}\parallel }^{2}}𝑑z\to -\mathrm{\infty }.$(3.38)

From (3.30) and (3.33), we have

${\parallel D{u}_{n}^{+}\parallel }_{p}^{p}+\frac{p}{2}{\parallel D{u}_{n}^{+}\parallel }_{2}^{2}-{\int }_{\mathrm{\Omega }}\left[\frac{\lambda p}{q}{|{u}_{n}^{+}|}^{q}+pF\left(z,{u}_{n}^{+}\right)\right]𝑑z\ge -{M}_{3}$(3.39)

for some ${M}_{3}>0$ and all $n\in ℕ$ (see (3.21)). Similarly, if we use (3.33) in (3.32) and also choose $h={u}_{n}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, then

$-{\parallel D{u}_{n}^{+}\parallel }_{p}^{p}-{\parallel D{u}_{n}^{+}\parallel }_{2}^{2}+{\int }_{\mathrm{\Omega }}\left[\lambda {\left({u}_{n}^{+}\right)}^{q}+f\left(z,{u}_{n}^{+}\right){u}_{n}^{+}\right]𝑑z\ge -{M}_{4}$(3.40)

for some ${M}_{4}>0$ and all $n\in ℕ$ (see (3.21)). We add (3.39) and (3.40) and obtain

$\left(\frac{p}{2}-1\right){\parallel D{u}_{n}^{+}\parallel }_{2}^{2}+{\int }_{\mathrm{\Omega }}\left[f\left(z,{u}_{n}^{+}\right){u}_{n}^{+}-pF\left(z,{u}_{n}^{+}\right)\right]𝑑z\ge -{M}_{5}+\lambda \left(\frac{p}{q}-1\right){\parallel {u}_{n}^{+}\parallel }_{q}^{q},$

with ${M}_{5}={M}_{3}+{M}_{4}>0$, for all $n\in ℕ$, hence

$\left(\frac{p}{2}-1\right){\parallel D{y}_{n}\parallel }_{2}^{2}+{\int }_{\mathrm{\Omega }}\frac{f\left(z,{u}_{n}^{+}\right){u}_{n}^{+}-pF\left(z,{u}_{n}^{+}\right)}{{\parallel {u}_{n}^{+}\parallel }^{2}}𝑑z\ge -\frac{{M}_{5}}{{\parallel {u}_{n}^{+}\parallel }^{2}}+\lambda \left(\frac{p}{q}-1\right)\frac{{\parallel {y}_{n}\parallel }_{q}^{q}}{{\parallel {u}_{n}^{+}\parallel }^{2-q}}.$

Since $q<2, passing to the limit as $n\to \mathrm{\infty }$, we have a contradiction (see (3.38)). Therefore, ${\left\{{u}_{n}^{+}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded, and hence ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded (see (3.33)).

We may assume that

(3.41)

In (3.32) we choose $h={u}_{n}-u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, pass to the limit as $n\to \mathrm{\infty }$ and use (3.41). Then

$\underset{n\to \mathrm{\infty }}{lim}\left[〈{A}_{p}\left({u}_{n}\right),{u}_{n}-u〉+〈A\left({u}_{n}\right),{u}_{n}-u〉\right]=0,$

which implies (recall that $A\left(\cdot \right)$ is monotone)

$\underset{n\to \mathrm{\infty }}{lim sup}\left[〈{A}_{p}\left({u}_{n}\right),{u}_{n}-u〉+〈A\left(u\right),{u}_{n}-u〉\right]\le 0.$

Hence (see (3.41)),

$\underset{n\to \mathrm{\infty }}{lim sup}〈{A}_{p}\left({u}_{n}\right),{u}_{n}-u〉\le 0,$

and thus (see Proposition 2.2)

Therefore, ${\phi }_{\lambda }^{+}$ satisfies the C-conditions and this proves claim 3.

Then (3.26) and claims 2 and 3, permit the use of Theorem 2.1 (the mountain pass theorem). So, we can find $\stackrel{^}{u}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that $\stackrel{^}{u}\in {K}_{{\phi }_{\lambda }^{+}}\subseteq \left[{\stackrel{~}{u}}_{\lambda }\right)\cap \mathrm{int}{C}_{+}$ (see claim 1) and ${m}_{\lambda }^{+}\le {\phi }_{\lambda }^{+}\left(\stackrel{^}{u}\right)$ (see (3.26)). From (3.21) and (3.26) we infer that $\stackrel{^}{u}$ is a positive solution of (1.1) and $\stackrel{^}{u}\ne {u}_{0}$. As in the proof of Proposition 3.2, using Proposition 2.4, we can show that $\stackrel{^}{u}-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+}.$

(b) Let ${\stackrel{~}{v}}_{\lambda }=-{\stackrel{~}{u}}_{\lambda }\in -\mathrm{int}{C}_{+}$ be the unique negative solution of (3.1) (see Proposition 3.1). We consider the following truncation of the reaction in problem (3.1):

This is a Caratheodory function. Set ${R}_{\lambda }^{-}\left(z,x\right)={\int }_{0}^{x}{r}_{\lambda }^{-}\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-function ${\phi }_{\lambda }^{-}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

Working with ${\phi }_{\lambda }^{-}$ as in part (a), we see that for $\lambda \in \left(0,{\lambda }_{-}^{*}\right]$, we can find $\stackrel{^}{v}\in -\mathrm{int}{C}_{+}$, a second negative solution of (1.1), such that $\stackrel{^}{v}\ne {v}_{0}$ and ${\stackrel{~}{v}}_{\lambda }-v\in \mathrm{int}{C}_{+}$. ∎

We will show that we have extremal constant sign solutions, that is, there is a smallest positive solution for problem (1.1), with $\lambda \in \left(0,{\lambda }_{+}^{*}\right]$, and a biggest negative solution for (1.1), with $\lambda \in \left(0,{\lambda }_{-}^{*}\right]$. These extremal constant sign solutions will be used in the next section to generate nodal solutions.

We introduce the following solution sets:

• ${S}_{\lambda }^{+}$ is the set of positive solutions for problem (1.1),

• ${S}_{\lambda }^{-}$ is the set of negative solutions for problem (1.1).

From Proposition 3.2 we know that

#### Proposition 3.4.

If Hypothesis 2.6 holds, then

• (a)

for all $\lambda \in \left(0,{\lambda }_{+}^{*}\right]$ , problem ( 1.1 ) has a smallest positive solution ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ , that is, ${\overline{u}}_{\lambda }\le u$ for all $u\in {S}_{\lambda }^{+}$,

• (b)

for all $\lambda \in \left(0,{\lambda }_{-}^{*}\right]$ , problem ( 1.1 ) has a biggest negative solution ${\overline{v}}_{\lambda }\in -\mathrm{int}{C}_{+}$ , that is, $v\le {\overline{v}}_{\lambda }$ for all $v\in {S}_{\lambda }^{-}$.

#### Proof.

(a) First we show that

(3.42)

To this end, let $u\in {S}_{\lambda }^{+}$ and consider the Caratheodory function

(3.43)

We set ${B}_{\lambda }\left(z,x\right)={\int }_{0}^{x}{\beta }_{\lambda }\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\stackrel{^}{\psi }}_{\lambda }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

From (3.43) it is clear that ${\stackrel{^}{\psi }}_{\lambda }\left(\cdot \right)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ${\stackrel{~}{u}}_{\lambda }^{*}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

${\stackrel{^}{\psi }}_{\lambda }\left({\stackrel{~}{u}}_{\lambda }^{*}\right)=inf\left[{\stackrel{^}{\psi }}_{\lambda }\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right].$(3.44)

Let $y\in \mathrm{int}{C}_{+}$. Since $u\in {S}_{\lambda }^{+}\subseteq \mathrm{int}{C}_{+}$, using [17, Proposition 2.1], we can find $t\in \left(0,1\right]$ small such that $ty\le u$. Then, from (3.43) and since $q<2, we see that by choosing $t\in \left(0,1\right]$ even smaller if necessary, we have ${\stackrel{^}{\psi }}_{\lambda }\left(ty\right)<0$, which implies ${\stackrel{^}{\psi }}_{\lambda }\left({\stackrel{~}{u}}_{\lambda }^{*}\right)<0={\stackrel{^}{\psi }}_{\lambda }\left(0\right)$ (see (3.44)), and thus ${\stackrel{~}{u}}_{\lambda }^{*}\ne 0$.

From (3.44) we have ${\stackrel{^}{\psi }}_{\lambda }^{\prime }\left({\stackrel{~}{u}}_{\lambda }^{*}\right)=0$, hence

(3.45)

In (3.45), we choose $h=-{\left({\stackrel{~}{u}}_{\lambda }^{*}\right)}^{-}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then ${\parallel D{\left({\stackrel{~}{u}}_{\lambda }^{*}\right)}^{-}\parallel }_{p}^{p}+{\parallel D{\left({\stackrel{~}{u}}_{\lambda }^{*}\right)}^{-}\parallel }_{2}^{2}=0$ (see (3.43)), and therefore ${\stackrel{~}{u}}_{\lambda }^{*}\ge 0$ and ${\stackrel{~}{u}}_{\lambda }^{*}\ne 0$.

Also, in (3.45), we choose $h={\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (see (3.12), (3.43), Hypothesis 2.6 (v) and note that $u\in {S}_{\lambda }^{+}$)

$〈{A}_{p}\left({\stackrel{~}{u}}_{\lambda }^{*}\right),{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}〉+〈A\left({\stackrel{~}{u}}_{\lambda }^{*}\right),{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}〉={\int }_{\mathrm{\Omega }}\lambda {u}^{q-1}{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}𝑑z$$\le {\int }_{\mathrm{\Omega }}\left[\lambda {u}^{q-1}+f\left(z,u\right)\right]{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}𝑑z$$=〈{A}_{p}\left(u\right),{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}〉+〈A\left(u\right),{\left({\stackrel{~}{u}}_{\lambda }^{*}-u\right)}^{+}〉,$

hence ${\stackrel{~}{u}}_{\lambda }^{*}\le u$.

We have proved that

${\stackrel{~}{u}}_{\lambda }^{*}\in \left[0,u\right],{\stackrel{~}{u}}_{\lambda }^{*}\ne 0.$(3.46)

From (3.46), (3.45), (3.43) and Proposition 3.1, we conclude that ${\stackrel{~}{u}}_{\lambda }^{*}={\stackrel{~}{u}}_{\lambda }$. Therefore, (3.42) is true.

From [8], we know that ${S}_{\lambda }^{+}$ is downward directed (that is, if ${u}_{1},{u}_{2}\in {S}_{\lambda }^{+}$, then we can find $v\in {S}_{\lambda }^{+}$ such that $u\le {u}_{1},u\le {u}_{2}$). Invoking [13, Lemma 3.10, p. 178], we can find a decreasing sequence ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {S}_{\lambda }^{+}$ such that $inf{S}_{\lambda }^{+}={inf}_{n\ge 1}{u}_{n}$. Evidently, ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded (see Hypothesis 2.6 (i) and recall that $0\le {u}_{n}\le {u}_{1}$ for all $n\in ℕ$). So, we may assume that

(3.47)

For every $n\in ℕ$, we have

(3.48)

In (3.48), we choose $h={u}_{n}-{\overline{u}}_{\lambda }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, pass to the limit as $n\to \mathrm{\infty }$, use (3.47), and reasoning as in the proof of Proposition 3.3, via Proposition 2.2, we conclude that

(3.49)

So, in (3.48), we pass to the limit as $n\to \mathrm{\infty }$ and use (3.49). Then

Also, from (3.42) and (3.49), we have ${\stackrel{~}{u}}_{\lambda }\le {\overline{u}}_{\lambda }$. Therefore,

(b) Similarly for the set ${S}_{\lambda }^{-}$, we have $v\le {\stackrel{~}{v}}_{\lambda }$ for all $v\in {S}_{\lambda }^{-}$. Reasoning as in part (a), we produce a ${\overline{v}}_{\lambda }\in {S}_{\lambda }^{-}$ such that $v\le {\overline{v}}_{\lambda }$ for all $v\in {S}_{\lambda }^{-}$. ∎

Next we examine the maps $\lambda ↦{\overline{u}}_{\lambda }$ and $\lambda ↦{\overline{v}}_{\lambda }$.

#### Proposition 3.5.

If Hypothesis 2.6 holds, then

• (a)

the map $\lambda ↦{\overline{u}}_{\lambda }$ from $\left(0,{\lambda }_{+}^{*}\right]$ into ${C}_{+}$ is strictly increasing (that is, for $0<\theta <\lambda \le {\lambda }_{+}^{*}$, ${\overline{u}}_{\lambda }-{\overline{u}}_{\theta }\in \mathrm{int}{C}_{+}$ ) and left continuous;

• (b)

the map $\lambda ↦{\overline{v}}_{\lambda }$ from $\left(0,{\lambda }_{-}^{*}\right]$ into $-{C}_{+}$ is strictly decreasing (that is, for $0<\theta <\lambda \le {\lambda }_{-}^{*}$, ${\overline{u}}_{\theta }-{\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ ) and left continuous.

#### Proof.

(a) First we show that $\lambda ↦{\overline{u}}_{\lambda }$ is increasing. We consider the Caratheodory function

(3.50)

We set ${\mathrm{\Gamma }}_{\theta }\left(z,x\right)={\int }_{0}^{x}{m}_{\theta }\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\sigma }_{\theta }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

This is coercive (see (3.50)) and sequentially weakly lower semicontinuous. Hence, we can find ${y}_{\theta }\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

${\sigma }_{\theta }\left({y}_{\theta }\right)=inf\left[{\sigma }_{\theta }\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right].$(3.51)

Exploiting as before the fact that $q<2, we have ${\sigma }_{\theta }\left({y}_{\theta }\right)<0={\sigma }_{\theta }\left(0\right)$. Hence, ${y}_{\theta }\ne 0$. From (3.51) we have ${\sigma }_{\theta }^{\prime }\left({y}_{\theta }\right)=0$, therefore

(3.52)

In (3.52), first let $h=-{y}_{\theta }^{-}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then, using (3.50), we have ${y}_{\theta }\ge 0$, ${y}_{\theta }\ne 0$. Also, in (3.52), we choose $h={\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then (see (3.50) and note that $\theta <\lambda$)

$〈{A}_{p}\left({y}_{\theta }\right),{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}〉+〈A\left({y}_{\theta }\right),{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}〉={\int }_{\mathrm{\Omega }}\left[\theta {\overline{u}}_{\lambda }^{q-1}+f\left(z,{\overline{u}}_{\lambda }\right)\right]{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}𝑑z$$\le {\int }_{\mathrm{\Omega }}\left[\lambda {\overline{u}}_{\lambda }^{q-1}+f\left(z,{\overline{u}}_{\lambda }\right)\right]{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}𝑑z$$=〈{A}_{p}\left({\overline{u}}_{\lambda }\right),{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}〉+〈A\left({\overline{u}}_{\lambda }\right),{\left({y}_{\theta }-{\overline{u}}_{\lambda }\right)}^{+}〉,$

hence ${y}_{\theta }\le {\overline{u}}_{\lambda }$.

We have proved that

${y}_{\theta }\in \left[0,{\overline{u}}_{\lambda }\right],{y}_{\theta }\ne 0,$(3.53)

which implies ${y}_{\theta }\in {S}_{\theta }^{+}\subseteq \mathrm{int}{C}_{+}$ (see (3.50), (3.52)), and thus ${\overline{u}}_{\theta }\le {y}_{\theta }\le {\overline{u}}_{\lambda }$ (see (3.53)). Hence, $\lambda ↦{\overline{u}}_{\lambda }$ is increasing. Now let $\rho ={\parallel {\overline{u}}_{\lambda }\parallel }_{\mathrm{\infty }}$ and let ${\stackrel{^}{\xi }}_{\rho }>0$ be as postulated by Hypothesis 2.6 (vi). Then we have (see Hypothesis 2.6 (vi) and recall that $\theta <\lambda$)

$-{\mathrm{\Delta }}_{p}{\overline{u}}_{\theta }-\mathrm{\Delta }{\overline{u}}_{\theta }+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\theta }^{p-1}=\theta {\overline{u}}_{\theta }^{q-1}+f\left(z,{\overline{u}}_{\theta }\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\theta }^{p-1}$$=\lambda {\overline{u}}_{\theta }^{q-1}+f\left(z,{\overline{u}}_{\theta }\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\theta }^{p-1}-\left(\lambda -\theta \right){\overline{u}}_{\theta }^{q-1}$$\le \lambda {\overline{u}}_{\theta }^{q-1}+f\left(z,{\overline{u}}_{\theta }\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\theta }^{p-1}$

Let

${h}_{1}\left(z\right)=\theta {\overline{u}}_{\theta }{\left(z\right)}^{q-1}+f\left(z,{\overline{u}}_{\theta }\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\theta }{\left(z\right)}^{p-1},$${h}_{2}\left(z\right)=\lambda {\overline{u}}_{\lambda }{\left(z\right)}^{q-1}+f\left(z,{\overline{u}}_{\lambda }\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\lambda }{\left(z\right)}^{p-1}.$

Evidently, ${h}_{1},{h}_{2}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and $\left(\lambda -\theta \right){\overline{u}}_{\theta }{\left(z\right)}^{q-1}\le {h}_{2}\left(z\right)-{h}_{1}\left(z\right)$ for a.a. $z\in \mathrm{\Omega }$. Since ${\overline{u}}_{\theta }\in \mathrm{int}{C}_{+}$, it follows that ${h}_{1}\prec {h}_{2}$ and so, using Proposition 2.4, we conclude that ${\overline{u}}_{\lambda }-{\overline{u}}_{\theta }\in \mathrm{int}{C}_{+}$. We have proved that $\lambda ↦{\overline{u}}_{\lambda }$ is strictly increasing.

Next we show that the map $\lambda ↦{\overline{u}}_{\lambda }$ is left continuous. To this end, let ${\lambda }_{n}\to {\lambda }^{-},\lambda \le {\lambda }_{+}^{*}$. For each $n\in ℕ$, let ${\overline{u}}_{n}={\overline{u}}_{{\lambda }_{n}}\in {S}_{{\lambda }_{n}}^{+}\subseteq \mathrm{int}{C}_{+}$ be the minimal positive solution of problem (1.1) (see Proposition 3.3). Then we have

$〈{A}_{p}\left({\overline{u}}_{n}\right),h〉+〈A\left({\overline{u}}_{n}\right),h〉={\int }_{\mathrm{\Omega }}\left[{\lambda }_{n}{\overline{u}}_{n}^{q-1}+f\left(z,{\overline{u}}_{n}\right)\right]h𝑑z$(3.54)

for all $h\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, all $n\in ℕ$, and

(3.55)

In (3.54), we choose $h={\overline{u}}_{n}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. Then from Hypothesis 2.6 (i) and (3.55) we infer that ${\left\{{\overline{u}}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded. Using [18, Corollary 8.6, p. 208], we can find ${c}_{6}>0$ such that

Invoking [15, Theorem 1], we can find $\alpha \in \left(0,1\right)$ and ${c}_{7}>0$ such that

Since ${C}_{0}^{1,\alpha }\left(\overline{\mathrm{\Omega }}\right)↪{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ compactly, we may assume that

(3.56)

Passing to the limit as $n\to \mathrm{\infty }$ in (3.54) and using (3.56), we obtain ${\stackrel{^}{u}}_{\lambda }\in {S}_{\lambda }^{+}$. We claim that ${\overline{u}}_{\lambda }={\stackrel{^}{u}}_{\lambda }$. If this is not true, then we can find ${z}_{0}\in \mathrm{\Omega }$ such that ${\overline{u}}_{\lambda }\left({z}_{0}\right)<{\stackrel{^}{u}}_{\lambda }\left({z}_{0}\right)$, hence (see (3.56))

This contradicts the fact that $\lambda ↦{\overline{u}}_{\lambda }$ is increasing. Hence, ${\stackrel{^}{u}}_{\lambda }={\overline{u}}_{\lambda }$ and so we conclude the left continuity of the map $\lambda ↦{\overline{u}}_{\lambda }$. ∎

Summarizing the results obtained in this section, we can state the following theorem.

#### Theorem 3.6.

If Hypothesis 2.6 holds, then

• (a)

there exists ${\lambda }_{+}^{*}>0$ such that for $\lambda \in \left(0,{\lambda }_{+}^{*}\right]$ , we have

• (i)(a)

problem ( 1.1 ) has at least two positive solutions

${u}_{0},\stackrel{^}{u}\in \mathrm{int}{C}_{+},{u}_{0}\ne \stackrel{^}{u},{u}_{0}-{\stackrel{~}{u}}_{\lambda },\stackrel{^}{u}-{\stackrel{~}{u}}_{\lambda }\in \mathrm{int}{C}_{+},$

• (i)(b)

problem ( 1.1 ) has a smallest positive solution ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ and $\lambda ↦{\overline{u}}_{\lambda }$ is strictly increasing and left continuous;

• (b)

there exists ${\lambda }_{-}^{*}>0$ such that for $\lambda \in \left(0,{\lambda }_{-}^{*}\right]$ , we have

• (ii)(a)

problem ( 1.1 ) has at least two negative solutions

${v}_{0},\stackrel{^}{v}\in -\mathrm{int}{C}_{+},{v}_{0}\ne \stackrel{^}{v},{\stackrel{~}{v}}_{\lambda }-{v}_{0},{\stackrel{~}{v}}_{\lambda }-\stackrel{^}{v}\in \mathrm{int}{C}_{+},$

• (ii)(b)

problem ( 1.1 ) has a biggest negative solution ${\overline{v}}_{\lambda }\in -\mathrm{int}{C}_{+}$ and $\lambda ↦{\overline{v}}_{\lambda }$ is strictly decreasing and left continuous.

## 4 Nodal solutions

In this section we focus on nodal (sign changing) solutions for problem (1.1). So, let ${\lambda }^{*}=\mathrm{min}\left\{{\lambda }_{+}^{*},{\lambda }_{-}^{*}\right\}$ (see Theorem 3.6). Let $\lambda \in \left(0,{\lambda }^{*}\right]$ and consider the ${C}^{1}$-functional ${\stackrel{^}{\tau }}_{\lambda }:{H}_{0}^{1}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

Hypothesis 2.6 (v) and Proposition 2.1 of [14] imply the following result.

#### Proposition 4.1.

If Hypothesis 2.6 holds and $\lambda \mathrm{>}\mathrm{0}$, then ${C}_{k}\mathit{}\mathrm{\left(}{\stackrel{\mathrm{^}}{\tau }}_{\lambda }\mathrm{,}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $k\mathrm{\in }{\mathrm{N}}_{\mathrm{0}}$.

Let ${\tau }_{\lambda }={{\stackrel{^}{\tau }}_{\lambda }|}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)}$. Since ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)↪{H}_{0}^{1}\left(\mathrm{\Omega }\right)$ densely, we can apply [19, Theorem 16] (see also [4, p. 14]) and have ${C}_{k}\left({\tau }_{\lambda },0\right)={C}_{k}\left({\stackrel{^}{\tau }}_{\lambda },0\right)$ for all $k\in {ℕ}_{0}$, thus (see [14])

(4.1)

Let ${\phi }_{\lambda }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ be the energy functional for problem (1.1) defined by

Then we have

$|{\phi }_{\lambda }\left(u\right)-{\tau }_{\lambda }\left(u\right)|=\frac{1}{p}{\parallel Du\parallel }_{p}^{p}=\frac{1}{p}{\parallel u\parallel }^{p},$

Therefore,

${\parallel {\phi }_{\lambda }^{\prime }\left(u\right)-{\tau }_{\lambda }^{\prime }\left(u\right)\parallel }_{*}\le {\parallel Du\parallel }_{p}^{p-1}={\parallel u\parallel }^{p-1}.$

Then the ${C}^{1}$-continuity property of critical groups (see [11, Theorem 5.126, p. 836]) implies that

hence (see (4.1))

(4.2)

Let $\lambda \in \left(0,{\lambda }^{*}\right]$ and let ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+},{\overline{v}}_{\lambda }\in -\mathrm{int}{C}_{+}$ be two extremal constant sign solutions for problem (1.1) (see Theorem 3.6). We introduce the following trunction of the reaction in problem (1.1):

(4.3)

This is a Caratheodory function. Set ${G}_{\lambda }\left(z,x\right)={\int }_{0}^{x}{g}_{\lambda }\left(z,s\right)𝑑s$ and consider the ${C}^{1}$-functional ${\stackrel{~}{\phi }}_{\lambda }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

We know that ${\stackrel{~}{\phi }}_{\lambda }\in {C}^{1}\left({W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$. Using (4.3), we can easily show that

${K}_{{\stackrel{~}{\phi }}_{\lambda }}\subseteq \left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right).$(4.4)

Therefore, the nontrivial critical points of ${\stackrel{~}{\phi }}_{\lambda }$, distinct from ${\overline{v}}_{\lambda }$ and ${\overline{u}}_{\lambda }$, are nodal solutions of (1.1). So, we assume that ${K}_{{\stackrel{~}{\phi }}_{\lambda }}$ is finite. Otherwise, we already have an infinity of nodal solutions for problem (1.1).

We also consider the positive and negative truncations of ${\stackrel{~}{\phi }}_{\lambda }$, namely, we consider the ${C}^{1}$-functionals ${\stackrel{~}{\phi }}_{\lambda }^{±}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to ℝ$ defined by

We can easily show that

${K}_{{\stackrel{~}{\phi }}_{\lambda }^{+}}\subseteq \left[0,{\overline{u}}_{\lambda }\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right),{K}_{{\stackrel{~}{\phi }}_{\lambda }^{-}}\subseteq \left[{\overline{v}}_{\lambda },0\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right).$

The extremality of ${\overline{u}}_{\lambda }$ and ${\overline{v}}_{\lambda }$ implies that

${K}_{{\stackrel{~}{\phi }}_{\lambda }^{+}}=\left\{0,{\overline{u}}_{\lambda }\right\},{K}_{{\stackrel{~}{\phi }}_{\lambda }^{-}}=\left\{0,{\overline{v}}_{\lambda }\right\}.$(4.5)

We compute the critical groups of ${\stackrel{~}{\phi }}_{\lambda }$.

#### Proposition 4.2.

If Hypothesis 2.6 holds and $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\lambda }^{\mathrm{*}}\mathrm{\right]}$, then ${C}_{k}\mathit{}\mathrm{\left(}{\phi }_{\lambda }\mathrm{,}\mathrm{0}\mathrm{\right)}\mathrm{=}{C}_{k}\mathit{}\mathrm{\left(}{\stackrel{\mathrm{~}}{\phi }}_{\lambda }\mathrm{,}\mathrm{0}\mathrm{\right)}$ for all $k\mathrm{\in }{\mathrm{N}}_{\mathrm{0}}$.

#### Proof.

We consider the homotopy $\stackrel{^}{h}\left(t,u\right)$ defined by

Suppose we can find ${\left\{{t}_{n}\right\}}_{n\ge 1}\subseteq \left[0,1\right]$ and ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

(4.6)

Then we have

$〈{A}_{p}\left({u}_{n}\right),h〉+〈A\left({u}_{n}\right),h〉=\left(1-{t}_{n}\right){\int }_{\mathrm{\Omega }}\left[\lambda {|{u}_{n}|}^{q-2}{u}_{n}+f\left(z,{u}_{n}\right)\right]h𝑑z+{t}_{n}{\int }_{\mathrm{\Omega }}{g}_{\lambda }\left(z,{u}_{n}\right)h𝑑z$

for all $h\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ and all $n\in ℕ$, which implies

(4.7)

Evidently, ${\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is bounded (see (4.6), (4.7)). Then as before using (4.7) and [15, Theorem 1], we can find $\alpha \in \left(0,1\right)$ and ${c}_{8}>0$ such that

(4.8)

From (4.6), (4.8) and the compact embedding of ${C}_{0}^{1,\alpha }\left(\overline{\mathrm{\Omega }}\right)$ into ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$, we have

hence

and therefore ${\left\{{u}_{n}\right\}}_{n\ge {n}_{0}}\subseteq {K}_{{\stackrel{~}{\phi }}_{\lambda }}$ (see (4.3)), a contradiction to our assumption that ${K}_{{\stackrel{~}{\phi }}_{\lambda }}$ is finite. Therefore, (4.6) can not occur, and from homotopy invariance of critical groups (see [11, Theorem 5.125, p. 836]), we have ${C}_{k}\left({\phi }_{\lambda },0\right)={C}_{k}\left({\stackrel{~}{\phi }}_{\lambda },0\right)$ for all $k\in {ℕ}_{0}$. ∎

From Proposition 4.2 and (4.2), we infer that

(4.9)

#### Proposition 4.3.

If Hypothesis 2.6 holds and $\mathrm{0}\mathrm{<}\lambda \mathrm{\le }{\lambda }^{\mathrm{*}}$, then problem (1.1) admits two nodal solutions

${y}_{0},\stackrel{^}{y}\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right),{y}_{0}\in \underset{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}{inf}\left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right].$

#### Proof.

From (4.4) and (4.5) we know that

${K}_{{\stackrel{~}{\phi }}_{\lambda }}\subseteq \left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right),{K}_{{\stackrel{~}{\phi }}_{\lambda }^{+}}=\left\{0,{\overline{u}}_{\lambda }\right\},{K}_{{\stackrel{~}{\phi }}_{\lambda }^{-}}=\left\{0,{\overline{v}}_{\lambda }\right\}.$(4.10)

Claim: ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+}$ and ${\overline{v}}_{\lambda }\in -\mathrm{int}{C}_{+}$ are local minimizers of ${\stackrel{~}{\phi }}_{\lambda }$.

From (4.3) it is clear that ${\stackrel{~}{\phi }}_{\lambda }^{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ${\overline{u}}_{\lambda }^{*}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that

${\stackrel{~}{\phi }}_{\lambda }^{+}\left({\overline{u}}_{\lambda }^{*}\right)=inf\left[{\stackrel{~}{\phi }}_{\lambda }^{+}\left(u\right):u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right].$(4.11)

As before, since $q<2, we have ${\stackrel{~}{\phi }}_{\lambda }^{+}\left({\overline{u}}_{\lambda }^{*}\right)<0={\stackrel{~}{\phi }}_{\lambda }^{+}\left(0\right)$, hence ${\overline{u}}_{\lambda }^{*}\ne 0$ and ${\overline{u}}_{\lambda }^{*}\in {K}_{{\stackrel{~}{\phi }}_{\lambda }^{+}}$ (see (4.11)), and thus ${\overline{u}}_{\lambda }^{*}={\overline{u}}_{\lambda }$ (see (4.10)). It is clear from (4.3) that ${{\stackrel{~}{\phi }}_{\lambda }|}_{{C}_{+}}={{\stackrel{~}{\phi }}_{\lambda }^{+}|}_{{C}_{+}}$. Since ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+}$, it follows that ${\overline{u}}_{\lambda }$ is a local ${C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$-minimizer of ${\stackrel{~}{\phi }}_{\lambda }$, hence ${\overline{u}}_{\lambda }$ is a local ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$-minimizer of ${\stackrel{~}{\phi }}_{\lambda }$ (see Proposition 2.3). We can argue in a similar manner for ${\overline{v}}_{\lambda }\in -\mathrm{int}{C}_{+}$, using this time the functional ${\stackrel{~}{\phi }}_{\lambda }^{-}$. This proves the claim.

Without loss of generality, we may assume that

${\stackrel{~}{\phi }}_{\lambda }\left({\overline{v}}_{\lambda }\right)\le {\stackrel{~}{\phi }}_{\lambda }\left({\overline{u}}_{\lambda }\right).$(4.12)

Recall that ${K}_{{\stackrel{~}{\phi }}_{\lambda }}$ is finite. So, on account of the claim and (4.12), we can find $\rho \in \left(0,1\right)$ small such that

${\stackrel{~}{\phi }}_{\lambda }\left({\overline{v}}_{\lambda }\right)\le {\stackrel{~}{\phi }}_{\lambda }\left({\overline{u}}_{\lambda }\right)\rho .$(4.13)

We know that ${\stackrel{~}{\phi }}_{\lambda }$ is coercive (see (4.3)). Therefore, we have that

(4.14)

(see [17, Proposition 2.2]). From (4.13), (4.14) we see that we can use Theorem 2.1 (the mountain pass theorem). Therefore, we can find ${y}_{0}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that ${y}_{0}\in {K}_{{\stackrel{~}{\phi }}_{\lambda }}\subseteq \left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right]\cap {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ (see (4.10)) and ${\stackrel{~}{m}}_{\lambda }\le {\stackrel{~}{\phi }}_{\lambda }\left({y}_{0}\right)$ (see (4.13)), thus ${y}_{0}\notin \left\{{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right\}$ (see (4.13)). Since ${y}_{0}$ is a critical point of mountain pass type for ${\stackrel{~}{\phi }}_{\lambda }$, we have

${C}_{1}\left({\stackrel{~}{\phi }}_{\lambda },{y}_{0}\right)\ne 0$(4.15)

(see [18, Proposition 6.100, p. 176]). From (4.9) and (4.15), we infer that ${y}_{0}\ne 0$, hence ${y}_{0}\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$ is a nodal solution of (1.1).

Let $a\left(y\right)={|y|}^{p-2}y+y$ for all $y\in {ℝ}^{N}$. Evidently, $a\in C\left({ℝ}^{N},{ℝ}^{N}\right)$ (recall that $2) and

We have

hence

(4.16)

We know that

Using (4.16) and the tangency principle of Pucci and Serrin [26, p. 35], we have

(4.17)

Let $\rho =\mathrm{max}\left\{{\parallel {\overline{u}}_{\lambda }\parallel }_{\mathrm{\infty }},{\parallel {\overline{v}}_{\lambda }\parallel }_{\mathrm{\infty }}\right\}$ and let ${\stackrel{^}{\xi }}_{\rho }>0$ be as postulated by Hypothesis 2.6 (vi). Then (see (4.17) and Hypothesis 2.6 (vi))

$-{\mathrm{\Delta }}_{p}{y}_{0}-\mathrm{\Delta }{y}_{0}+{\stackrel{^}{\xi }}_{\rho }{|{y}_{0}|}^{p-2}{y}_{0}=\lambda {|{y}_{0}|}^{p-2}{y}_{0}+f\left(z,{y}_{0}\right)+{\stackrel{^}{\xi }}_{\rho }{|{y}_{0}|}^{p-2}{y}_{0}$$\le \lambda {\overline{u}}_{\lambda }^{q-1}+f\left(z,{\overline{u}}_{\lambda }\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\lambda }^{p-1}$(4.18)

Let

${h}_{1}\left(z\right)=\lambda {|{y}_{0}\left(z\right)|}^{q-2}{y}_{0}\left(z\right)+f\left(z,{y}_{0}\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{|{y}_{0}\left(z\right)|}^{p-2}{y}_{0}\left(z\right),$${h}_{2}\left(z\right)=\lambda {\overline{u}}_{\lambda }{\left(z\right)}^{q-1}+f\left(z,{\overline{u}}_{\lambda }\left(z\right)\right)+{\stackrel{^}{\xi }}_{\rho }{\overline{u}}_{\lambda }{\left(z\right)}^{p-1}.$

Evidently, ${h}_{1},{h}_{2}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and

${h}_{2}\left(z\right)-{h}_{1}\left(z\right)\ge \lambda \left[{u}_{\lambda }{\left(z\right)}^{q-1}-{|{y}_{0}\left(z\right)|}^{q-2}{y}_{0}\left(z\right)\right],$

which implies ${h}_{1}\prec {h}_{2}$ (see Hypothesis 2.6 (vi), (4.17) and recall ${\overline{u}}_{\lambda }\in \mathrm{int}{C}_{+},{y}_{0}\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$).

Then from (4.18) and Proposition 2.4, we infer that

${\overline{u}}_{\lambda }-{y}_{0}\in \mathrm{int}{C}_{+}.$

Similarly, we show that

${y}_{0}-{\overline{v}}_{\lambda }\in \mathrm{int}{C}_{+}.$

Therefore,

${y}_{0}\in {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right].$(4.19)

On the other hand, from [12, Proposition 17] (see also [20]), we have a nodal solution of (1.1) such that

$\stackrel{^}{y}\notin {\mathrm{int}}_{{C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)}\left[{\overline{v}}_{\lambda },{\overline{u}}_{\lambda }\right].$(4.20)

From (4.19) and (4.20) it follows that ${y}_{0}\ne \stackrel{^}{y}$. ∎

We can state the following multiplicity result for problem (1.1).

#### Theorem 4.4.

If Hypothesis 2.6 holds, then there exists ${\lambda }^{\mathrm{*}}\mathrm{>}\mathrm{0}$ such that for all $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\lambda }^{\mathrm{*}}\mathrm{\right]}$, problem (1.1) has at least six nontrivial smooth solutions

${u}_{0},\stackrel{^}{u}\in \mathrm{int}{C}_{+},$${u}_{0}\ne \stackrel{^}{u},$${v}_{0},\stackrel{^}{v}\in -\mathrm{int}{C}_{+},$${v}_{0}\ne \stackrel{^}{v},$${y}_{0},\stackrel{^}{y}\in {C}_{0}^{1}\left(\overline{\mathrm{\Omega }}\right)$$\mathit{\text{both nodal}}.$

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Accepted: 2018-08-16

Published Online: 2018-10-11

Published in Print: 2019-03-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11671111

Funding Source: Natural Science Foundation of Heilongjiang Province

Award identifier / Grant number: LBHQ16082

This work was supported by the NSFC (No. 11671111) and Heilongjiang Province Postdoctoral Startup Foundation (Grant No. LBHQ16082).

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 228–249, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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