# Existence of solutions for nonlinear Robin problems with the p-Laplacian and hemivariational inequality

## Abstract

In this paper, we show the existence of at least three nontrivial solutions for a nonlinear elliptic equation driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality) and Robin boundary condition. Two of these solutions are of constant sign (one is positive, the other negative). We mainly use a variational approach together with a sub-sup solution method.

## 1 Introduction

Consider the problem

$\left\{\begin{array}{ll}âˆ’{\mathrm{Î”}}_{p}x+\mathrm{Î±}{|x|}^{pâˆ’2}xâˆˆ\mathrm{âˆ‚}j\left(z,x\right),& zâˆˆZ,\\ {|\mathrm{âˆ‡}x|}^{pâˆ’2}\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}+b\left(z\right){|x|}^{pâˆ’2}x=0,& zâˆˆ\mathrm{âˆ‚}Z,\end{array}$
(1.1)

where $ZâŠ‚{\mathbb{R}}^{N}$ is a bounded domain with ${C}^{2}$-boundary âˆ‚Z, ${\mathrm{Î”}}_{p}x=\text{div}\left({|\mathrm{âˆ‡}x|}^{pâˆ’2}\mathrm{âˆ‡}x\right)$ ($1) is the p-Laplacian operator, $\mathrm{Î±}>0$, $b\left(z\right)âˆˆ{L}^{\mathrm{âˆž}}\left(\mathrm{âˆ‚}Z\right)$, $b\left(z\right)â‰¥0$, and on âˆ‚Z. $j\left(z,x\right)$ is a measurable potential function on $ZÃ—\mathbb{R}$, which is locally Lipschitz in the $xâˆˆ\mathbb{R}$, $\mathrm{âˆ‚}j\left(z,x\right)$ stands for the generalized subdifferential of $xâ†¦j\left(z,x\right)$. Also $\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}$ denotes the outer normal derivative of x with respect to âˆ‚Z. The aim of this paper is to prove the existence of two constant sign solutions and furthermore prove the existence of at least three nontrivial solutions for problem (1.1).

A multiplicity of solutions for problems driven by the p-Laplacian has been obtained by Ambrosetti et al.[1] and Garcia Azorero et al.[2]. In these works, the authors deal with a right-hand side nonlinearity of the form $âˆ’{\mathrm{Î”}}_{p}x=\mathrm{Î»}{|x|}^{qâˆ’2}x+{|x|}^{râˆ’2}x$ with $\mathrm{Î»}>0$ being a real parameter, $1 (${p}^{âˆ—}=\frac{Np}{Nâˆ’p}$ if $p; ${p}^{âˆ—}=+\mathrm{âˆž}$ otherwise) and prove the existence of positive and negative solutions. The question of the existence of a p-Laplacian Robin problem $âˆ’{\mathrm{Î”}}_{p}x+\mathrm{Î±}{|x|}^{pâˆ’2}x=j\left(z,x\right)$ was also present in the work of Zhang et al.[3] for $p=2$, the authors show that the Robin problem has at least four nontrivial solutions using a sub-sup solution method, the FucÃ­k spectrum, the mountain pass theorem, and the degree theorem together. In the work of Zhang et al.[4], [5] for $p>2$, the authors show that the oscillating equations with the p-Laplacian Robin problem has infinitely many nontrivial solutions. In Anello [6] and Ricceri [7], the main tool is an abstract variational principle of Ricceri and its use is made possible by the hypothesis that $p>N$; by the fact that Sobolev space ${W}^{1,p}\left(Z\right)$ is compactly embedded in $C\left(\stackrel{Â¯}{Z}\right)$, the authors obtain infinitely many weak solutions for p-Laplacian Neumann problem.

In all the aforementioned works, the nonlinearity is a CarathÃ©odory function, a.e. $j\left(z,x\right)$ is continuously differentiable in the variable x. In Barletta and Papageorgiou [8], the authors consider a nonsmooth potential with an asymmetric behavior at +âˆž and at âˆ’âˆž to get two nontrivial solutions using degree methods. Also, in Dancer and Du [9], the authors use the critical point theory and a sub-sup solution method on smooth critical point theory.

In this paper, we use a combination of nonsmooth critical point theory with sub-sup solution methods. We also use the nonsmooth version of the second deformation theorem due to Corvellec [10]. Thus, we can extend the works of [9], [11]â€“[13] to a hemivariational inequality with the Robin boundary condition.

## 2 Preliminaries

Now we recall the subdifferential theory for locally Lipschitz functions and the corresponding nonsmooth critical point theory. Let X be a Banach space and let ${X}^{âˆ—}$ be its topological dual. We denote by $ã€ˆâ‹\dots ,â‹\dots ã€‰$ the duality brackets for the pair $\left(X,{X}^{âˆ—}\right)$. The generalized directional derivative ${\mathrm{Ï†}}^{0}\left(x;h\right)$ of a locally Lipschitz function $\mathrm{Ï†}:Xâ†’\mathbb{R}$ at $xâˆˆX$ along the direction $hâˆˆX$ is defined as follows:

${\mathrm{Ï†}}^{0}\left(x;h\right)=\underset{yâ†’x,\mathrm{Î»}â†’0}{limâ€‰sup}\frac{\mathrm{Ï†}\left(y+\mathrm{Î»}h\right)âˆ’\mathrm{Ï†}\left(y\right)}{\mathrm{Î»}}.$

It is well known that ${\mathrm{Ï†}}^{0}\left(x;â‹\dots \right)$ is sublinear continuous and it is the support function of a nonempty, convex, and ${w}^{âˆ—}$-compact set $\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)âŠ†{X}^{âˆ—}$ defined by

$\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)=\left\{{x}^{âˆ—}âˆˆ{X}^{âˆ—}:ã€ˆ{x}^{âˆ—},hã€‰â‰¤{\mathrm{Ï†}}^{0}\left(x;h\right),\mathrm{âˆ€}hâˆˆX\right\}.$

The function $\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)$ is the â€˜generalized subdifferentialâ€™ of Ï†. If $\mathrm{Ï†}âˆˆ{C}^{1}\left(X\right)$, then Ï† is locally Lipschitz and $\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)=\left\{{\mathrm{Ï†}}^{â€²}\left(x\right)\right\}$. Moreover, if Ï† is also convex, then $\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)$ coincides with the subdifferential in the sense of convex analysis, ${\mathrm{âˆ‚}}_{c}\mathrm{Ï†}\left(x\right)$, which is defined by

${\mathrm{âˆ‚}}_{c}\mathrm{Ï†}\left(x\right)=\left\{{x}^{âˆ—}âˆˆ{X}^{âˆ—}:ã€ˆ{x}^{âˆ—},hã€‰â‰¤\mathrm{Ï†}\left(x+h\right)âˆ’\mathrm{Ï†}\left(x\right),\mathrm{âˆ€}hâˆˆX\right\}.$

If $0âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)$, then we call $xâˆˆX$ critical point of Ï†. It is easy to see that if $xâˆˆX$ is a local minimum or a local maximum of Ï†, then $xâˆˆX$ is a critical point of Ï†.

A locally Lipschitz function Ï† satisfies the Palais-Smale condition at level $câˆˆ\mathbb{R}$, if every sequence ${\left\{{x}_{n}\right\}}_{nâ‰¥1}âŠ†X$ satisfying $\mathrm{Ï†}\left({x}_{n}\right)â†’c$ and $inf\left\{âˆ¥{x}^{âˆ—}âˆ¥:{x}^{âˆ—}âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left({x}_{n}\right)\right\}â†’0$ as $nâ†’\mathrm{âˆž}$ has a strongly convergent subsequence. If Ï† satisfies the Palais-Smale condition at level $câˆˆ\mathbb{R}$ for all $câˆˆ\mathbb{R}$, then we say that it satisfies the Palais-Smale condition. For the details, we refer to [14].

In the following study, denote $R\left(z,x\right)={|\mathrm{âˆ‡}x|}^{pâˆ’2}\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}+b\left(z\right){|x|}^{pâˆ’2}x$, and we will use the following spaces:

Both are ordered Banach spaces, and we denote

It is well known that the principal eigenfunction $eâˆˆint\left({C}^{+}\right)$, so .

Furthermore, define ${u}_{1}$ as the normalized principal eigenfunction of $\left(âˆ’{\mathrm{Î”}}_{p},{W}_{n}^{1,p}\left(Z\right)\right)$ (see [15]). It is well known that ${u}_{1}\left(z\right)â‰¥0$, a.e. $zâˆˆZ$, from the nonlinear regularity ${u}_{1}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$ (see Di Benedetto [16], [17], Chapter IX]), furthermore ${u}_{1}âˆˆint\left({C}^{+}\right)$ by virtue of the strong maximum principle of Vazquez [18].

We give the following minimax characterization (see [19]), suited for our purpose.

### Proposition 2.1

Let$S={W}_{n}^{1,p}\left(Z\right)âˆ©\mathrm{âˆ‚}{B}_{1}$and$\mathrm{Î“}=\left\{\mathrm{Î³}âˆˆC\left(\left[0,1\right],S\right):\mathrm{Î³}\left(0\right)=âˆ’{u}_{1},\mathrm{Î³}\left(1\right)={u}_{1}\right\}$, where$\mathrm{âˆ‚}{B}_{1}=\left\{xâˆˆ{L}^{p}\left(Z\right):{âˆ¥xâˆ¥}_{p}=1\right\}$. Then the first eigenvalue${\mathrm{Î»}}_{1}$of$\left(âˆ’{\mathrm{Î”}}_{p},{W}_{n}^{1,p}\left(Z\right)\right)$equals

${\mathrm{Î»}}_{1}=\underset{\mathrm{Î³}âˆˆ\mathrm{Î“}}{inf}\underset{tâˆˆ\left[0,1\right]}{max}{âˆ¥D\mathrm{Î³}\left(t\right)âˆ¥}_{p}^{p}.$

Next we recall the definitions of sub-sup solutions for problem (1.1).

1. (1)

A function $\stackrel{Â¯}{x}âˆˆ{W}^{1,p}\left(Z\right)$ with $R\left(z,\stackrel{Â¯}{x}\left(z\right)\right)â‰¥0$ is called a â€˜sup solutionâ€™, if

$\begin{array}{r}{âˆ«}_{Z}{|D\stackrel{Â¯}{x}\left(z\right)|}^{pâˆ’2}\left(D\stackrel{Â¯}{x}\left(z\right),Dy\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{Z}{|\stackrel{Â¯}{x}\left(z\right)|}^{pâˆ’2}\left(\stackrel{Â¯}{x}\left(z\right),y\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|\stackrel{Â¯}{x}\left(z\right)|}^{pâˆ’2}\left(\stackrel{Â¯}{x}\left(z\right),y\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dzâ‰¥{âˆ«}_{Z}u\left(z\right)y\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\end{array}$

for all $yâˆˆ{W}_{n}^{1,p}\left(Z\right)$, $y\left(z\right)â‰¥0$ a.e. on Z and for some $uâˆˆ{L}^{q}\left(Z\right)$, $u\left(z\right)âˆˆ\mathrm{âˆ‚}j\left(z,\stackrel{Â¯}{x}\left(z\right)\right)$ a.e. on Z for some $1 if $N>p$, $q=+\mathrm{âˆž}$ if $Nâ‰¤p$.

1. (2)

A function $\underset{Ì²}{x}âˆˆ{W}^{1,p}\left(Z\right)$ with $R\left(z,\underset{Ì²}{x}\left(z\right)\right)â‰¤0$ is called a â€˜sub-solutionâ€™, if

$\begin{array}{r}{âˆ«}_{Z}{|D\underset{Ì²}{x}\left(z\right)|}^{pâˆ’2}\left(D\underset{Ì²}{x}\left(z\right),Dy\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{Z}{|\underset{Ì²}{x}\left(z\right)|}^{pâˆ’2}\left(\underset{Ì²}{x}\left(z\right),y\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|\underset{Ì²}{x}\left(z\right)|}^{pâˆ’2}\left(\underset{Ì²}{x}\left(z\right),y\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dzâ‰¤{âˆ«}_{Z}u\left(z\right)y\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\end{array}$

for all $yâˆˆ{W}_{n}^{1,p}\left(Z\right)$, $y\left(z\right)â‰¥0$ a.e. on Z and for some $uâˆˆ{L}^{q}\left(Z\right)$, $u\left(z\right)âˆˆ\mathrm{âˆ‚}j\left(z,\underset{Ì²}{x}\left(z\right)\right)$ a.e. on Z for some $1 if $N>p$, $q=+\mathrm{âˆž}$ if $Nâ‰¤p$.

Finally we recall the following topological notion which is crucial in critical point theory.

### Definition 2.2

[20]

Let S, Q be closed subsets of a Banach space X, Q with relative boundary âˆ‚Q. We say S and âˆ‚Q link if

1. (1)

$Sâˆ©\mathrm{âˆ‚}Q=\mathrm{âˆ\dots }$, and

2. (2)

for any map $hâˆˆ{C}^{0}\left(X,X\right)$ such that $h{|}_{\mathrm{âˆ‚}Q}=id$ we have .

From the definition, we give the following general minimax principle for the critical values of a locally Lipschitz function Ï†.

### Proposition 2.3

[20]

Suppose Ï† is locally Lipschitz and satisfies the$\left(PS\right)$-condition. Consider closed subsets$S,QâŠ‚X$and Q with relative boundary âˆ‚Q. Suppose

1. (1)

2. (2)

${inf}_{S}\mathrm{Ï†}>{sup}_{\mathrm{âˆ‚}Q}\mathrm{Ï†}$.

Let

$\mathrm{Î“}=\left\{hâˆˆ{C}^{0}\left(X,X\right):h{|}_{\mathrm{âˆ‚}Q}=id\right\}.$

Then the number

$\mathrm{Î²}=\underset{hâˆˆ\mathrm{Î“}}{inf}\underset{uâˆˆQ}{sup}\mathrm{Ï†}\left(h\left(u\right)\right)$

defines a critical value$\mathrm{Î²}â‰¥{inf}_{S}\mathrm{Ï†}$of Ï†.

### Remark 2.4

From the above general minimax principle, a nonsmooth version of the mountain pass theorem, the saddle point theorem, and the generalized mountain pass theorem are available by choosing the link sets appropriately (see [10], [14]).

The following result is the so-called â€˜second deformation theoremâ€™ for a nonsmooth setting. In fact, this result is due to Corvellec [10]. We give the following sets:

$\begin{array}{c}K=\left\{xâˆˆX:0âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right)\right\},\hfill \\ {K}_{c}=\left\{xâˆˆX:0âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left(x\right),\mathrm{Ï†}\left(x\right)=c\right\},\hfill \\ {\mathrm{Ï†}}^{c}=\left\{xâˆˆX:\mathrm{Ï†}\left(x\right)

We know that K, ${K}_{c}$, and ${\mathrm{Ï†}}^{c}$ are the critical set of Ï†, the critical set at level $câˆˆ\mathbb{R}$ of Ï†, and the strict sublevel set of Ï† at c, respectively.

### Proposition 2.5

Let X be a Banach space, $\mathrm{Ï†}:Xâ†’\mathbb{R}$be locally Lipschitz satisfying the Palais-Smale condition. $a,bâˆˆ\mathbb{R}$with$a. Assume also that$Kâˆ©{\mathrm{Ï†}}^{âˆ’1}\left(\left(a,b\right)\right)=\mathrm{âˆ\dots }$and${K}_{a}$is a finite set containing only local minimizers of Ï†.

Then there exists a continuous deformation $\mathrm{Î¦}:\left[0,1\right]Ã—{\mathrm{Ï†}}^{b}â†’{\mathrm{Ï†}}^{b}$ such that

1. (1)

$\mathrm{Î¦}\left(t,x\right)=x$ for all $tâˆˆ\left[0,1\right]$, $xâˆˆ{K}_{a}$,

2. (2)

$\mathrm{Î¦}\left(1,{\mathrm{Ï†}}^{b}\right)âŠ†{\mathrm{Ï†}}^{a}âˆª{K}_{a}$,

3. (3)

$\mathrm{Ï†}\left(\mathrm{Î¦}\left(t,x\right)\right)â‰¤\mathrm{Ï†}\left(x\right)$ for all $tâˆˆ\left[0,1\right]$, $xâˆˆ{\mathrm{Ï†}}^{b}$.

### Definition 2.6

[21]

Let X be a topological space and A a subspace of X. A weak deformation retraction from X to A is a homology $F:XÃ—Iâ†’X$ such that for all $xâˆˆX$ and $aâˆˆA$, we have $F\left(x,0\right)=x$, $F\left(a,1\right)=a$, and $F\left(x,1\right)âˆˆA$.

In particular, the set ${\mathrm{Ï†}}^{a}âˆª{K}_{a}$ is a weak deformation retract of ${\mathrm{Ï†}}^{b}$.

We now recall another notion, which will be useful in the following. Suppose W is a Banach space and $A:Wâ†’{W}^{âˆ—}$ is a mapping, we say that A is a type ${\left(S\right)}_{+}$ if for every sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{âˆž}}$ such that ${x}_{n}â‡€xâˆˆW$ and ${limâ€‰sup}_{nâ†’\mathrm{âˆž}}ã€ˆA\left({x}_{n}\right),{x}_{n}âˆ’xã€‰â‰¤0$.

Considering the nonlinear mapping $A:{W}_{n}^{1,p}\left(Z\right)â†’{W}_{n}^{1,p}{\left(Z\right)}^{âˆ—}$ defined for all $xâˆˆ{W}_{n}^{1,p}\left(Z\right)$ by

$ã€ˆA\left(x\right),yã€‰={âˆ«}_{Z}{|Dx\left(z\right)|}^{pâˆ’2}Dx\left(z\right)â‹\dots Dy\left(z\right)\phantom{\rule{0.2em}{0ex}}dz.$
(2.1)

We have the following result (see [8], Proposition 4.1]).

### Proposition 2.7

The mapping (2.1) is continuous and of the type${\left(S\right)}_{+}$.

### Definition 2.8

[14]

Given a functional $\mathrm{Ï†}:{W}_{n}^{1,p}\left(Z\right)â†’\mathbb{R}$, ${x}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ is called a W-local minimizer of Ï† if there exists $r>0$ satisfying for all $yâˆˆ{W}_{n}^{1,p}\left(Z\right)$ with ${âˆ¥yâˆ¥}_{{W}_{n}^{1,p}\left(Z\right)}â‰¤r$, we have

$\mathrm{Ï†}\left({x}_{0}\right)â‰¤\mathrm{Ï†}\left({x}_{0}+y\right).$

### Definition 2.9

[14]

${x}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ is called a C-local minimizer of Ï† if there exists $r>0$ satisfying for all $yâˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$ with ${âˆ¥yâˆ¥}_{{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)}â‰¤r$, we have

$\mathrm{Ï†}\left({x}_{0}\right)â‰¤\mathrm{Ï†}\left({x}_{0}+y\right).$

As the study of problems like (1.1) is reduced to seeking the critical points of corresponding energy functional on ${W}_{n}^{1,p}\left(Z\right)$ or on ${C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$, in this section we introduce the notations used along the paper together with the main abstract results that we will use later on for a C-local minimizer to be a W-local minimizer. Such a result for $p>2$ was first proved in [2]. Then it has been extended to the Neumann boundary condition and a nonsmooth potential by [8].

We denote $\mathrm{Ïˆ}:{W}_{n}^{1,p}\left(Z\right)â†’\mathbb{R}$ for all $xâˆˆ{W}_{n}^{1,p}\left(Z\right)$

$\mathrm{Ïˆ}\left(x\right)=\frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}j\left(z,x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz.$

From Clarke [22], pp.32-34], we know that Ïˆ is locally Lipschitz. By [12], we know that if we let ${x}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ be a C-local minimizer of Ïˆ, then ${x}_{0}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$ and it is a W-local minimizer of Ïˆ.

## 3 Solutions of constant sign

In this section, by using a sub-sup solution method, we get two solutions of (1.1) with constant sign, one positive and the other negative.

Our general assumptions on the nonsmooth potential $j\left(z,x\right)$ are the following:

$A\left(j\right)$:

1. (i)

$zâ†¦j\left(z,x\right)$ is measurable for all $xâˆˆ\mathbb{R}$;

2. (ii)

$xâ†¦j\left(z,x\right)$ is locally Lipschitz for a.e. $zâˆˆZ$;

3. (iii)

$|u|â‰¤\mathrm{Î³}\left(z\right)+C{|x|}^{pâˆ’1}$ for a.e. $zâˆˆZ$, all $xâˆˆ\mathbb{R}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Î³}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ and $C>0$;

4. (iv)

${limâ€‰sup}_{|x|â†’\mathrm{âˆž}}\frac{u}{{|x|}^{pâˆ’2}x}â‰¤\mathrm{Ï‰}\left(z\right)$ for a.e. $zâˆˆZ$, all $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Ï‰}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ satisfying $\mathrm{Ï‰}\left(z\right)â‰¤\mathrm{Î±}$ a.e. in Z and $\mathrm{Ï‰}\left(z\right)<\mathrm{Î±}$ in some set of positive measure;

5. (v)

$\mathrm{Î·}\left(z\right)+\mathrm{Î±}â‰¤{limâ€‰inf}_{xâ†’0}\frac{u}{{|x|}^{pâˆ’2}x}$ for a.e. $zâˆˆZ$, all $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Î·}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ satisfying ${\mathrm{Î»}}_{1}â‰¤\mathrm{Î·}\left(z\right)$ a.e. in Z and ${\mathrm{Î»}}_{1}<\mathrm{Î·}\left(z\right)$ in some set of positive measure, ${\mathrm{Î»}}_{1}$ is the first eigenvalue of $âˆ’{\mathrm{Î”}}_{p}$ with Robin boundary condition;

6. (vi)

$uxâ‰¥0$ for a.e. $zâˆˆZ$, all $xâˆˆ\mathbb{R}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$.

### Theorem 3.1

Assume that$A\left(j\right)$(i)-(vi) hold. Problem (1.1) has at least two solutions${x}_{0}âˆˆint\left({C}_{+}\right)$and${x}_{âˆ—}âˆˆâˆ’int\left({C}_{+}\right)$.

### Example

The following potential function j satisfies assumptions $A\left(j\right)$ (for the sake of simplicity we drop the z-dependence):

$j\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{Î·}+\mathrm{Î±}}{p}{|x|}^{p},& |x|â‰¤1,\\ \frac{\mathrm{Ï‰}}{p}{|x|}^{p}+Cln{|x|}^{p}+\frac{\mathrm{Î·}+\mathrm{Î±}âˆ’\mathrm{Ï‰}}{p},& |x|>1,\end{array}$

where $0<\mathrm{Ï‰}<\mathrm{Î±}$, $\mathrm{Î·}>{\mathrm{Î»}}_{1}$, and $C>0$. Note that, if $C=\frac{\mathrm{Î·}+\mathrm{Î±}âˆ’\mathrm{Ï‰}}{p}$, then $jâˆˆ{C}^{1}\left(\mathbb{R}\right)$.

Note that

$\mathrm{âˆ‚}j\left(x\right)=\left\{\begin{array}{cc}\left[âˆ’\mathrm{Ï‰}âˆ’pC,âˆ’\mathrm{Î·}âˆ’\mathrm{Î±}\right],\hfill & x=âˆ’1,\hfill \\ \left(\mathrm{Î·}+\mathrm{Î±}\right){|x|}^{pâˆ’2}x,\hfill & |x|â‰¤1,\hfill \\ 0,\hfill & x=0,\hfill \\ \mathrm{Ï‰}{|x|}^{pâˆ’2}x+pC\frac{x}{{|x|}^{2}},\hfill & |x|>1,\hfill \\ \left[\mathrm{Î·}+\mathrm{Î±},\mathrm{Ï‰}+pC\right],\hfill & x=1.\hfill \end{array}$

It is easy to see that j satisfies $A\left(j\right)$(i)-(iii), (vi). For all $uâˆˆ\mathrm{âˆ‚}j\left(x\right)$, we have

$\underset{|x|â†’\mathrm{âˆž}}{limâ€‰sup}\frac{u}{{|x|}^{pâˆ’2}x}â‰¤\mathrm{Ï‰}$

and

$\mathrm{Î·}+\mathrm{Î±}â‰¤\underset{xâ†’0}{limâ€‰inf}\frac{u}{{|x|}^{pâˆ’2}x}.$

Then the potential function j satisfies assumptions $A\left(j\right)$.

### Remark 3.2

In fact, problem (1.1) has the trivial solution $0âˆˆ\mathrm{âˆ‚}j\left(z,0\right)$ for a.e. $zâˆˆZ$ according to assumption $A\left(j\right)$(vi) and the upper semicontinuity of the subdifferential $\mathrm{âˆ‚}j\left(z,â‹\dots \right)$ (see Clarke [22]). What we are interesting in is whether it has nontrivial solutions.

We introduce a useful extension of the notion of maximal monotonicity (see [14], p.83]).

### Definition 3.3

Let X be a reflexive Banach space and $A:Xâ†’{2}^{{X}^{âˆ—}}$ an operator. We say that A is pseudomonotone if

1. (1)

A has nonempty, bounded and convex values;

2. (2)

A is upper semicontinuous for every finite dimensional subspace of X into ${X}^{âˆ—}$;

3. (3)

if ${x}_{n}â‡€x$ in X, ${x}_{n}^{âˆ—}âˆˆA\left({x}_{n}\right)$, and ${limâ€‰sup}_{nâ†’+\mathrm{âˆž}}{ã€ˆ{x}_{n}^{âˆ—},{x}_{n}âˆ’xã€‰}_{X}â‰¤0$, then for every $yâˆˆX$, there exists ${u}^{âˆ—}\left(y\right)âˆˆA\left(x\right)$, such that

${ã€ˆ{u}^{âˆ—}\left(y\right),xâˆ’yã€‰}_{X}â‰¤\underset{nâ†’+\mathrm{âˆž}}{limâ€‰inf}{ã€ˆ{x}_{n}^{âˆ—},{x}_{n}âˆ’yã€‰}_{X}.$

### Definition 3.4

[14]

A is said to be demicontinuous on X if $\left\{{x}_{n}\right\}âŠ‚X$ and ${x}_{n}â†’xâˆˆX$ together imply $A\left({x}_{n}\right)â‡€A\left(x\right)$.

It is well known that (1.1) is the Euler-Lagrange equation of the functional $\mathrm{Ï†}:{W}_{n}^{1,p}\left(Z\right)â†’\mathbb{R}$,

$\mathrm{Ï†}\left(x\right)=\frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}j\left(z,x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆ{W}_{n}^{1,p}\left(Z\right).$

We introduce the truncation function ${\mathrm{Î½}}_{+}:\mathbb{R}â†’{\mathbb{R}}_{+}$ by

${\mathrm{Î½}}_{+}\left(x\right)=\left\{\begin{array}{ll}x,& x>0,\\ 0,& xâ‰¤0;\end{array}$

then define the locally Lipschitz functional ${\mathrm{Ï†}}_{+}:{W}_{n}^{1,p}\left(Z\right)â†’\mathbb{R}$ by

${\mathrm{Ï†}}_{+}\left(x\right)=\frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}{j}_{+}\left(z,x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆ{W}_{n}^{1,p}\left(Z\right),$

where ${j}_{+}\left(z,x\right)=j\left(z,{\mathrm{Î½}}_{+}\left(x\right)\right)$ for all $zâˆˆ\mathbb{R}$, $xâˆˆ\mathbb{R}$, which is locally Lipschitz.

We consider the nonlinear Robin problem for given $\mathrm{Îµ}>0$ and ${\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right)âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$, :

$\left\{\begin{array}{ll}âˆ’{\mathrm{Î”}}_{p}x+\mathrm{Î±}{|x|}^{pâˆ’2}x=\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){|x|}^{pâˆ’2}x+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right),& zâˆˆZ,\\ {|\mathrm{âˆ‡}x|}^{pâˆ’2}\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}+b\left(z\right){|x|}^{pâˆ’2}x=0,& zâˆˆ\mathrm{âˆ‚}Z.\end{array}$
(3.1)

Define the mapping $I:{W}_{n}^{1,p}\left(Z\right)â†’{W}_{n}^{1,p}{\left(Z\right)}^{âˆ—}$ for all $x,yâˆˆ{W}_{n}^{1,p}\left(Z\right)$ by

$\begin{array}{rcl}ã€ˆI\left(x\right),yã€‰& =& {âˆ«}_{Z}{|Dx\left(z\right)|}^{pâˆ’2}Dx\left(z\right)â‹\dots Dy\left(z\right)\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{Z}{|x\left(z\right)|}^{pâˆ’2}x\left(z\right)â‹\dots y\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ +{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x\left(z\right)|}^{pâˆ’2}x\left(z\right)â‹\dots y\left(z\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

It is well known that I is strictly monotone and demicontinuous, furthermore, maximal monotone (see [23]). We denote ${K}_{\mathrm{Îµ}}:{L}^{p}\left(Z\right)â†’{L}^{{p}^{â€²}}\left(Z\right)$ ($\frac{1}{p}+\frac{1}{{p}^{â€²}}=1$) and we have

${K}_{\mathrm{Îµ}}\left(x\right)\left(â‹\dots \right)=\left(\mathrm{Ï‰}\left(â‹\dots \right)+\mathrm{Îµ}\right){|x\left(â‹\dots \right)|}^{pâˆ’2}x\left(â‹\dots \right),$

which is bounded and continuous. Then the mapping $I\left(x\right)âˆ’{K}_{\mathrm{Îµ}}\left(x\right)$ is pseudomonotone from ${W}_{n}^{1,p}\left(Z\right)$ into ${W}_{n}^{âˆ’1,{p}^{â€²}}\left(Z\right)$, in fact, ${W}_{n}^{1,p}\left(Z\right)â†ª{L}^{p}\left(Z\right)$ is compact embedding and ${K}_{\mathrm{Îµ}}:{W}_{n}^{1,p}\left(Z\right)â†’{L}^{{p}^{â€²}}\left(Z\right)$ is completely continuous.

Next, we will show that (3.1) has a solution $\stackrel{Â¯}{x}âˆˆint\left({C}_{+}\right)$.

### Lemma 3.5

Let$\mathrm{Ï‰}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$satisfy$\mathrm{Ï‰}\left(z\right)â‰¤\mathrm{Î±}$a.e. in Z and$\mathrm{Ï‰}\left(z\right)<\mathrm{Î±}$in some set of positive measure. Then (3.1) has a solution$\stackrel{Â¯}{x}âˆˆint\left({C}_{+}\right)$for$\mathrm{Îµ}>0$small enough.

### Proof

First, we claim that there exists ${\mathrm{Î¾}}_{0}>0$ such that

$J\left(x\right)={âˆ¥Dxâˆ¥}_{p}^{p}+\mathrm{Î±}{âˆ¥xâˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâ‰¥{\mathrm{Î¾}}_{0}{âˆ¥xâˆ¥}_{p}^{p},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆ{W}_{n}^{1,p}\left(Z\right).$

In fact, from assumption $bâ‰¥0$, we know that $J\left(x\right)â‰¥0$, for all $xâˆˆ{W}_{n}^{1,p}\left(Z\right)$. Suppose the conclusion is false, we have ${x}_{n}âˆˆ{W}_{n}^{1,p}\left(Z\right)$, $J\left({x}_{n}\right)<\frac{1}{n}{âˆ¥{x}_{n}âˆ¥}_{p}^{p}$, . If we set ${x}_{n}^{â€²}=\frac{{x}_{n}}{âˆ¥{x}_{n}âˆ¥}$, then $J\left({x}_{n}^{â€²}\right)<\frac{1}{n}$ (J is p-homogeneous). We may assume ${x}_{n}^{â€²}â‡€x$ in ${W}_{n}^{1,p}\left(Z\right)$, ${x}_{n}^{â€²}â†’x$ in ${L}^{p}\left(Z\right)$ by passing to a subsequence if necessary. Then

$\begin{array}{c}{âˆ¥xâˆ¥}_{p}^{p}=\underset{nâ†’\mathrm{âˆž}}{lim}{âˆ¥{x}_{n}^{â€²}âˆ¥}_{p}^{p},\phantom{\rule{2em}{0ex}}{âˆ¥Dxâˆ¥}_{p}^{p}â‰¤\underset{nâ†’\mathrm{âˆž}}{limâ€‰inf}{âˆ¥D{x}_{n}^{â€²}âˆ¥}_{p}^{p},\hfill \\ {âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dz=\underset{nâ†’\mathrm{âˆž}}{lim}\left({âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{x}_{n}^{â€²}|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|{x}_{n}^{â€²}\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dz\right).\hfill \end{array}$

So, by passing to the limit of J, we have

${âˆ¥Dxâˆ¥}_{p}^{p}+\mathrm{Î±}{âˆ¥xâˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâ‰¤0.$

This implies

${âˆ¥Dxâˆ¥}_{p}^{p}â‰¤{âˆ«}_{Z}\left(\mathrm{Ï‰}\left(z\right)âˆ’\mathrm{Î±}\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâ‰¤0.$

Hence, we have $x\left(z\right)=C$ for a.e. $zâˆˆZ$ where $Câˆˆ\mathbb{R}$. In fact, $C=0$, if not, from the above inequality,

$0â‰¤{|C|}^{p}\left[{âˆ«}_{Z}\left(\mathrm{Ï‰}\left(z\right)âˆ’\mathrm{Î±}\right)\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right)\phantom{\rule{0.2em}{0ex}}ds\right]<0.$

It produces a contradiction. On the other hand,

${âˆ¥D{x}_{n}^{â€²}âˆ¥}_{p}^{p}=J\left({x}_{n}^{â€²}\right)âˆ’\mathrm{Î±}{âˆ¥{x}_{n}^{â€²}âˆ¥}_{p}^{p}âˆ’{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{x}_{n}^{â€²}|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|{x}_{n}^{â€²}\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dz.$

We have ${âˆ¥D{x}_{n}^{â€²}âˆ¥}_{p}â†’0$, together with ${x}_{n}â†’x$ in ${L}^{p}\left(Z\right)$, so ${x}_{n}â†’0$ in ${W}_{n}^{1,p}\left(Z\right)$, but ${âˆ¥{x}_{n}âˆ¥}_{{W}_{n}^{1,p}\left(Z\right)}=1$, $nâˆˆZ$. So the assumption is false, we have the conclusion.

For all $xâˆˆ{W}_{n}^{1,p}\left(Z\right)$, from the above discussion, we get

$\begin{array}{rcl}ã€ˆI\left(x\right)âˆ’{K}_{\mathrm{Îµ}}\left(x\right),xã€‰& =& {âˆ¥Dxâˆ¥}_{p}^{p}+\mathrm{Î±}{âˆ¥xâˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Îµ}{âˆ¥xâˆ¥}_{p}^{p}\\ â‰¥& \left({\mathrm{Îµ}}_{0}âˆ’\mathrm{Îµ}\right){âˆ¥xâˆ¥}_{p}^{p}.\end{array}$

So if $\mathrm{Îµ}<{\mathrm{Îµ}}_{0}$ small enough, we have $I\left(â‹\dots \right)âˆ’{K}_{\mathrm{Îµ}}\left(â‹\dots \right)$ is coercive. But a pseudomonotone coercive operator is surjective (see [23], Theorem 9.57]), for ${\mathrm{Î´}}_{\mathrm{Îµ}}$, we can find $\stackrel{Â¯}{x}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ such that

$I\left(\stackrel{Â¯}{x}\right)âˆ’{K}_{\mathrm{Îµ}}\left(\stackrel{Â¯}{x}\right)={\mathrm{Î´}}_{\mathrm{Îµ}}.$

That is,

$\left\{\begin{array}{l}âˆ’{\mathrm{Î”}}_{p}\stackrel{Â¯}{x}+\mathrm{Î±}{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}=\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right),\phantom{\rule{1em}{0ex}}zâˆˆZ,\\ {|\mathrm{âˆ‡}\stackrel{Â¯}{x}|}^{pâˆ’2}\frac{\mathrm{âˆ‚}\stackrel{Â¯}{x}}{\mathrm{âˆ‚}n}+b\left(z\right){|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}=0,\phantom{\rule{1em}{0ex}}zâˆˆ\mathrm{âˆ‚}Z.\end{array}$
(3.2)

It follows that $\stackrel{Â¯}{x}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ is a solution of (3.1).

Next we show $\stackrel{Â¯}{x}âˆˆint\left({C}_{+}\right)$. Take $âˆ’{\stackrel{Â¯}{x}}_{âˆ’}=âˆ’max\left\{âˆ’\stackrel{Â¯}{x},0\right\}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ for ${\mathrm{Î´}}_{\mathrm{Îµ}}â‰¥0$, then

$\begin{array}{rl}ã€ˆI\left(\stackrel{Â¯}{x}\right)âˆ’{K}_{\mathrm{Îµ}}\left(\stackrel{Â¯}{x}\right),âˆ’{\stackrel{Â¯}{x}}_{âˆ’}ã€‰=& âˆ’{âˆ¥D{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}âˆ’\mathrm{Î±}{âˆ¥{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}âˆ’{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{\stackrel{Â¯}{x}}_{âˆ’}|}^{p}\phantom{\rule{0.2em}{0ex}}ds\\ +{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|{\stackrel{Â¯}{x}}_{âˆ’}\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Îµ}{âˆ¥{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}â‰¥0.\end{array}$

So

${\mathrm{Îµ}}_{0}{âˆ¥{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}â‰¤{âˆ¥D{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}+\mathrm{Î±}{âˆ¥{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{\stackrel{Â¯}{x}}_{âˆ’}|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|{\stackrel{Â¯}{x}}_{âˆ’}\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâ‰¤\mathrm{Îµ}{âˆ¥{\stackrel{Â¯}{x}}_{âˆ’}âˆ¥}_{p}^{p}.$

But $\mathrm{Îµ}<{\mathrm{Îµ}}_{0}$, we have ${\stackrel{Â¯}{x}}_{âˆ’}=0$, that is, $\stackrel{Â¯}{x}â‰¥0$. Since ${\mathrm{Î´}}_{\mathrm{Îµ}}>0$, from (3.2), we have and $\stackrel{Â¯}{x}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$ (nonlinear regularity theorem, see [24]), furthermore, ${\mathrm{Î”}}_{p}\stackrel{Â¯}{x}â‰¤0$ on Z, so $\stackrel{Â¯}{x}âˆˆint\left({C}_{+}\right)$.â€ƒâ–¡

Now we prove that the solution $\stackrel{Â¯}{x}$ of (3.1) is a strict sup solution of (1.1) for $\mathrm{Îµ}>0$ small enough.

### Lemma 3.6

Let assumptions$A\left(j\right)$(i)-(iv) hold. Then the solution$\stackrel{Â¯}{x}$of (3.1) is a strict sup solution of (1.1) for$\mathrm{Îµ}>0$small enough.

### Proof

From $A\left(j\right)$(iv), for given $\mathrm{Îµ}>0$, we can find ${M}_{1}>0$, such that for all $zâˆˆZ$, $xâ‰¥{M}_{1}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, we have

$\frac{u}{{x}^{pâˆ’1}}â‰¤\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}.$

From $A\left(j\right)$(iii), we can find ${\mathrm{Î´}}_{\mathrm{Îµ}}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$, , such that for all $zâˆˆZ$, $xâˆˆ\left[0,{M}_{1}\right]$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, we have

$u<{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right).$

So for all $zâˆˆZ$, $xâ‰¥0$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, we have

$u<\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){x}^{pâˆ’1}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right).$

From Lemma 3.5, we see that (3.1) has a solution $\stackrel{Â¯}{x}âˆˆint\left({C}_{+}\right)$, so when $\mathrm{Îµ}<{\mathrm{Îµ}}_{0}$ small enough, for all $zâˆˆZ$, $xâˆˆ{L}^{{p}^{â€²}}{\left(Z\right)}_{+}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,\stackrel{Â¯}{x}\left(z\right)\right)$, we have

$u<\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){\stackrel{Â¯}{x}}^{pâˆ’1}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right),$

that is,

$u<âˆ’{\mathrm{Î”}}_{p}\stackrel{Â¯}{x}+\mathrm{Î±}{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x},$

and from the definition of a sup solution, we know that $\stackrel{Â¯}{x}$ is a sup solution of (1.1).â€ƒâ–¡

### Remark 3.7

We have found a sup solution of (1.1) and $\mathrm{âˆ‚}j\left(z,0\right)=\left\{0\right\}$ a.e. on Z, we also find $\underset{Ì²}{x}â‰¡0$ is a sub-solution of (1.1). Define the set

Next, we will find a nontrivial solution of (1.1) in W.

### Proof of Theorem 3.1

Step 1: Claim: We can find ${x}_{0}âˆˆW$ which is a local minimizer of ${\mathrm{Ï†}}_{+}$ and of Ï†.

From the discussion of Lemma 3.6, for a.e. $zâˆˆZ$, all $xâ‰¥0$, $uâˆˆ\mathrm{âˆ‚}{j}_{+}\left(z,x\right)=\mathrm{âˆ‚}j\left(z,x\right)$, we have

$u<\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){x}^{pâˆ’1}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right).$

Furthermore, for a.e. $zâˆˆZ$, all $xâ‰¥0$, from assumptions $A\left(j\right)$(i), (ii),

$\frac{d}{dx}{j}_{+}\left(z,x\right)<\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){x}^{pâˆ’1}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right)$

then for a.e. $zâˆˆZ$, all $xâ‰¥0$, we have

${j}_{+}\left(z,x\right)<\frac{1}{p}\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){|x|}^{p}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right)|x|.$

So, for some $C>0$, we have

$\begin{array}{rcl}{\mathrm{Ï†}}_{+}\left(x\right)& =& \frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}{j}_{+}\left(z,x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz\\ >& \frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’\frac{1}{p}{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\frac{\mathrm{Îµ}}{p}{âˆ¥xâˆ¥}_{p}^{p}âˆ’C{âˆ¥xâˆ¥}_{p}\\ â‰¥& \frac{1}{p}\left({\mathrm{Îµ}}_{0}âˆ’\mathrm{Îµ}\right){âˆ¥xâˆ¥}_{p}^{p}âˆ’C{âˆ¥xâˆ¥}_{p}.\end{array}$

Because of $\mathrm{Îµ}<{\mathrm{Îµ}}_{0}$, $p>1$, we see that ${\mathrm{Ï†}}_{+}$ is coercive, and together with ${\mathrm{Ï†}}_{+}$ weakly lower semicontinuous on W. Thus by the Weierstrass theorem, we can find ${x}_{0}âˆˆW$, satisfying

${\mathrm{Ï†}}_{+}\left({x}_{0}\right)=\underset{W}{inf}{\mathrm{Ï†}}_{+}.$

We claim that . In fact, from assumption $A\left(j\right)$(v), we see that, for given $\mathrm{Îµ}>0$, we can find some $\mathrm{Î´}>0$, for a.e. $zâˆˆZ$, all $xâˆˆ\left[0,\mathrm{Î´}\right]$, $uâˆˆ\mathrm{âˆ‚}{j}_{+}\left(z,x\right)=\mathrm{âˆ‚}j\left(z,x\right)$,

$\frac{u}{{x}^{pâˆ’1}}â‰¥\mathrm{Î·}\left(z\right)+\mathrm{Î±}âˆ’\mathrm{Îµ};$

then for a.e. $zâˆˆZ$ and all $xâˆˆ\left[0,\mathrm{Î´}\right]$, we get

${j}_{+}\left(z,x\right)â‰¥\frac{1}{p}\left(\mathrm{Î·}\left(z\right)+\mathrm{Î±}âˆ’\mathrm{Îµ}\right){x}^{p}.$

Furthermore, let ${e}_{1}$ be the first eigenfunction of Robin problem of $âˆ’{\mathrm{Î”}}_{p}$ (see [15]), then for $\stackrel{Â¯}{x}âˆˆint\left({C}^{+}\right)$, we can find $\mathrm{Î¸}>0$, such that

$\mathrm{Î¸}{e}_{1}\left(z\right)â‰¤min\left\{\stackrel{Â¯}{x}\left(z\right),\mathrm{Î´}\right\},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}zâˆˆ\stackrel{Â¯}{Z}.$

Then $\mathrm{Î¸}{e}_{1}âˆˆint\left({C}^{+}\right)$, and

$\begin{array}{rcl}{\mathrm{Ï†}}_{+}\left(\mathrm{Î¸}{e}_{1}\right)& =& \frac{{\mathrm{Î¸}}^{p}}{p}{âˆ¥D{e}_{1}âˆ¥}_{p}^{p}+\frac{\mathrm{Î±}{\mathrm{Î¸}}^{p}}{p}{âˆ¥{e}_{1}âˆ¥}_{p}^{p}+\frac{{\mathrm{Î¸}}^{p}}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}{j}_{+}\left(z,\mathrm{Î¸}{e}_{1}\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz\\ â‰¤& \frac{{\mathrm{Î¸}}^{p}}{p}{âˆ¥D{e}_{1}âˆ¥}_{p}^{p}âˆ’\frac{{\mathrm{Î¸}}^{p}}{p}{âˆ«}_{Z}\mathrm{Î·}\left(z\right){|{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dz+\frac{{\mathrm{Î¸}}^{p}}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}ds+\frac{\mathrm{Îµ}{\mathrm{Î¸}}^{p}}{p}{âˆ¥{e}_{1}âˆ¥}_{p}^{p}\\ =& \frac{{\mathrm{Î¸}}^{p}}{p}\left[{âˆ«}_{Z}\left({\mathrm{Î»}}_{1}âˆ’\mathrm{Î·}\left(z\right)\right){|{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Îµ}{âˆ¥{e}_{1}âˆ¥}_{p}^{p}\right].\end{array}$

From assumption $A\left(j\right)$(v) and ${e}_{1}>0$, we have

${âˆ«}_{Z}\left({\mathrm{Î»}}_{1}âˆ’\mathrm{Î·}\left(z\right)\right){|{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dz<0.$

If we choose Îµ small enough, we can get ${\mathrm{Ï†}}_{+}\left(\mathrm{Î¸}{e}_{1}\right)<0$ for all $\mathrm{Î¸}>0$ small enough. So, we have

${\mathrm{Ï†}}_{+}\left({x}_{0}\right)=\underset{W}{inf}{\mathrm{Ï†}}_{+}â‰¤{\mathrm{Ï†}}_{+}\left(\mathrm{Î¸}{e}_{1}\right)<0={\mathrm{Ï†}}_{+}\left(0\right),$

then we have , ${x}_{0}âˆˆW$.

Step 2: The local minimizer of ${\mathrm{Ï†}}_{+}$, ${x}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ is a nontrivial solution of (1.1).

Firstly, we claim that ${x}_{0}$ is also a local ${W}_{n}^{1,p}\left(Z\right)$-minimizer of ${\mathrm{Ï†}}_{+}$. In fact, the nonlinear regularity theory (see for example [24]) assures that ${x}_{0}âˆˆ{C}^{1}\left(\stackrel{Â¯}{Z}\right)$. Hence, as the boundary relation is understood in a pointwise sense and we get ${x}_{0}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$, also, by , ${x}_{0}â‰¥0$, and the nonlinear strong maximum principle of Vazquez, ${x}_{0}âˆˆint\left({C}_{+}\right)$, $\stackrel{Â¯}{x}âˆ’{x}_{0}âˆˆint\left({C}_{+}\right)$. So we can find $\mathrm{Î´}>0$ satisfying

$\begin{array}{c}{B}_{\mathrm{Î´}}\left({x}_{0}\right)=\left\{xâˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right):{âˆ¥xâˆ’{x}_{0}âˆ¥}_{{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)}<\mathrm{Î´}\right\}âŠ†int\left({C}_{+}\right),\hfill \\ {B}_{\mathrm{Î´}}\left(\stackrel{Â¯}{x}âˆ’{x}_{0}\right)=\left\{xâˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right):{âˆ¥xâˆ’\left(\stackrel{Â¯}{x}âˆ’{x}_{0}\right)âˆ¥}_{{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)}<\mathrm{Î´}\right\}âŠ†int\left({C}_{+}\right).\hfill \end{array}$

Then

${x}_{0}+{B}_{\mathrm{Î´}}âŠ†int\left({C}_{+}\right),\phantom{\rule{2em}{0ex}}\stackrel{Â¯}{x}âˆ’{x}_{0}+{B}_{\mathrm{Î´}}âŠ†int\left({C}_{+}\right).$

So, ${x}_{0}$ is also a local minimizer of ${\mathrm{Ï†}}_{+}$ on ${C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$; also from [24], ${x}_{0}$ is also a local ${W}_{n}^{1,p}\left(Z\right)$-minimizer of ${\mathrm{Ï†}}_{+}$ and of Ï† too.

Also, from [25], there exists $\mathrm{Ï‰}\left(z\right)âˆˆ\mathrm{âˆ‚}{j}_{+}\left(z,{x}_{0}\left(z\right)\right)=\mathrm{âˆ‚}j\left(z,{x}_{0}\left(z\right)\right)$, $uâˆˆ{L}^{{p}^{â€²}}\left(Z\right)$ satisfying

$0â‰¤ã€ˆI\left({x}_{0}\right),yâˆ’{x}_{0}ã€‰âˆ’{âˆ«}_{Z}u\left(z\right)\left(yâˆ’{x}_{0}\right)\left(z\right)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆW.$

Using

$y\left(z\right)=\left\{\begin{array}{cc}\stackrel{Â¯}{x}\left(z\right),\hfill & zâˆˆ\left\{\stackrel{Â¯}{x}â‰¤{x}_{0}+\mathrm{Îµ}v\right\}=A,\hfill \\ {x}_{0}\left(z\right)+\mathrm{Îµ}v\left(z\right),\hfill & zâˆˆ\left\{0<{x}_{0}+\mathrm{Îµ}v<\stackrel{Â¯}{x}\right\}=B,\hfill \\ 0,\hfill & zâˆˆ\left\{{x}_{0}+\mathrm{Îµ}vâ‰¤0\right\}=C.\hfill \end{array}$

We have $yâˆˆW$ for all $vâˆˆ{W}_{n}^{1,p}\left(Z\right)$, $\mathrm{Îµ}>0$, then we have

$\begin{array}{rcl}0& â‰¤& {âˆ«}_{A}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},D\left(\stackrel{Â¯}{x}âˆ’{x}_{0}\right)ã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{A}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},\stackrel{Â¯}{x}âˆ’{x}_{0}ã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{A}u\left(\stackrel{Â¯}{x}âˆ’{x}_{0}\right)\phantom{\rule{0.2em}{0ex}}dz\\ +\mathrm{Îµ}{âˆ«}_{B}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{B}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},\mathrm{Îµ}vã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{B}u\left(\mathrm{Îµ}v\right)\phantom{\rule{0.2em}{0ex}}dz\\ +{âˆ«}_{\mathrm{âˆ‚}B}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},\mathrm{Îµ}vã€‰\phantom{\rule{0.2em}{0ex}}ds\\ âˆ’{âˆ«}_{C}{|D{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Î±}{âˆ«}_{C}{|{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}dz+{âˆ«}_{C}u{x}_{0}\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{\mathrm{âˆ‚}C}b\left(z\right){|{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}ds\\ +{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},\stackrel{Â¯}{x}âˆ’{x}_{0}ã€‰\phantom{\rule{0.2em}{0ex}}ds\\ =& \mathrm{Îµ}{âˆ«}_{Z}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Îµ}\mathrm{Î±}{âˆ«}_{Z}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Îµ}{âˆ«}_{Z}uv\phantom{\rule{0.2em}{0ex}}dz\\ +\mathrm{Îµ}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}ds\\ âˆ’{âˆ«}_{C}{|D{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Î±}{âˆ«}_{C}{|{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Îµ}{âˆ«}_{C}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’{âˆ«}_{\mathrm{âˆ‚}C}b\left(z\right){|{x}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}ds\\ âˆ’{âˆ«}_{A}{|D\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆD\stackrel{Â¯}{x},D\left({x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}\right)ã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Î±}{âˆ«}_{A}{|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},{x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dz\\ +{âˆ«}_{A}\stackrel{Â¯}{u}\left({x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}\right)\phantom{\rule{0.2em}{0ex}}dz\\ âˆ’{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right){|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},{x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}ds+{âˆ«}_{C}u\left({x}_{0}+\mathrm{Îµ}v\right)\phantom{\rule{0.2em}{0ex}}dz+{âˆ«}_{A}\left(\stackrel{Â¯}{u}âˆ’u\right)\left(\stackrel{Â¯}{x}âˆ’{x}_{0}âˆ’\mathrm{Îµ}v\right)\phantom{\rule{0.2em}{0ex}}dz\\ +{âˆ«}_{A}ã€ˆ{|D\stackrel{Â¯}{x}|}^{pâˆ’2}D\stackrel{Â¯}{x}âˆ’{|D{x}_{0}|}^{pâˆ’2}D{x}_{0},D\left({x}_{0}âˆ’\stackrel{Â¯}{x}\right)ã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{A}ã€ˆ{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}âˆ’{|{x}_{0}|}^{pâˆ’2}{x}_{0},{x}_{0}âˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dz\\ +{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right)ã€ˆ{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}âˆ’{|{x}_{0}|}^{pâˆ’2}{x}_{0},{x}_{0}âˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}ds+\mathrm{Îµ}{âˆ«}_{C}ã€ˆ{|D\stackrel{Â¯}{x}|}^{pâˆ’2}D\stackrel{Â¯}{x}âˆ’{|D{x}_{0}|}^{pâˆ’2}D{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz\\ âˆ’\mathrm{Îµ}{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dsâˆ’\mathrm{Îµ}{âˆ«}_{\mathrm{âˆ‚}C}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}ds+\mathrm{Îµ}{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right){|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},vã€‰\phantom{\rule{0.2em}{0ex}}ds\\ âˆ’\mathrm{Îµ}\mathrm{Î±}{âˆ«}_{A}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dxâˆ’\mathrm{Îµ}\mathrm{Î±}{âˆ«}_{C}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dx+\mathrm{Îµ}\mathrm{Î±}{âˆ«}_{A}{|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},vã€‰\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

From the monotonicity of I, we have

$\begin{array}{r}{âˆ«}_{A}ã€ˆ{|D\stackrel{Â¯}{x}|}^{pâˆ’2}D\stackrel{Â¯}{x}âˆ’{|D{x}_{0}|}^{pâˆ’2}D{x}_{0},D\left({x}_{0}âˆ’\stackrel{Â¯}{x}\right)ã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Î±}{âˆ«}_{A}ã€ˆ{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}âˆ’{|{x}_{0}|}^{pâˆ’2}{x}_{0},{x}_{0}âˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}+{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right)ã€ˆ{|\stackrel{Â¯}{x}|}^{pâˆ’2}\stackrel{Â¯}{x}âˆ’{|{x}_{0}|}^{pâˆ’2}{x}_{0},{x}_{0}âˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dsâ‰¤0.\end{array}$

From the definition of a sup solution of (1.1), we have

$\begin{array}{r}âˆ’{âˆ«}_{A}{|D\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆD\stackrel{Â¯}{x},D\left({x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}\right)ã€‰\phantom{\rule{0.2em}{0ex}}dzâˆ’\mathrm{Î±}{âˆ«}_{A}{|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},{x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dz+{âˆ«}_{A}\stackrel{Â¯}{u}\left({x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}\right)\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}âˆ’{âˆ«}_{\mathrm{âˆ‚}A}b\left(z\right){|\stackrel{Â¯}{x}|}^{pâˆ’2}ã€ˆ\stackrel{Â¯}{x},{x}_{0}+\mathrm{Îµ}vâˆ’\stackrel{Â¯}{x}ã€‰\phantom{\rule{0.2em}{0ex}}dsâ‰¤0.\end{array}$

From $A\left(j\right)$(vi), we have ${âˆ«}_{C}u\left({x}_{0}+\mathrm{Îµ}v\right)\phantom{\rule{0.2em}{0ex}}dzâ‰¤0$. Furthermore,

${âˆ«}_{C}\left(\stackrel{Â¯}{u}âˆ’u\right)\left(\stackrel{Â¯}{x}âˆ’{x}_{0}âˆ’\mathrm{Îµ}v\right)\phantom{\rule{0.2em}{0ex}}dzâ‰¤\mathrm{Îµ}c{âˆ«}_{\left\{{x}_{0}+\mathrm{Îµ}vâ‰¥\stackrel{Â¯}{x}>{x}_{0}\right\}}v\phantom{\rule{0.2em}{0ex}}dz.$

Also,

and

So, we have

$\begin{array}{rcl}0& â‰¤& \mathrm{Îµ}{âˆ«}_{Z}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Îµ}\mathrm{Î±}{âˆ«}_{Z}{|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dz\\ +\mathrm{Îµ}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|{x}_{0}|}^{pâˆ’2}ã€ˆ{x}_{0},vã€‰\phantom{\rule{0.2em}{0ex}}dsâˆ’\mathrm{Îµ}{âˆ«}_{Z}uv\phantom{\rule{0.2em}{0ex}}dz\\ âˆ’\mathrm{Îµ}{âˆ«}_{C}{|D{x}_{0}|}^{pâˆ’2}ã€ˆD{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz+\mathrm{Îµ}c{âˆ«}_{\left\{{x}_{0}+\mathrm{Îµ}vâ‰¥\stackrel{Â¯}{x}>{x}_{0}\right\}}v\phantom{\rule{0.2em}{0ex}}dz\\ +\mathrm{Îµ}{âˆ«}_{C}ã€ˆ{|D\stackrel{Â¯}{x}|}^{pâˆ’2}D\stackrel{Â¯}{x}âˆ’{|D{x}_{0}|}^{pâˆ’2}D{x}_{0},Dvã€‰\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

As $\mathrm{Îµ}â†’0$, for all $vâˆˆ{W}_{n}^{1,p}\left(Z\right)$, we obtain

$0â‰¤ã€ˆI\left({x}_{0}\right),vã€‰âˆ’{âˆ«}_{Z}uv\phantom{\rule{0.2em}{0ex}}dz=ã€ˆI\left({x}_{0}\right)âˆ’u,vã€‰.$

That is,

$I\left({x}_{0}\right)=u.$

Then ${x}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ is a solution of (1.1).

Step 3: In a similar way, we introduce another truncation function ${\mathrm{Î½}}_{âˆ’}:\mathbb{R}â†’{\mathbb{R}}_{âˆ’}$ by

${\mathrm{Î½}}_{âˆ’}\left(x\right)=\left\{\begin{array}{ll}x,& x<0,\\ 0,& xâ‰¥0,\end{array}$

then define the locally Lipschitz functional ${\mathrm{Ï†}}_{âˆ’}:{W}_{n}^{1,p}\left(Z\right)â†’\mathbb{R}$ by

${\mathrm{Ï†}}_{âˆ’}\left(x\right)=\frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}{j}_{âˆ’}\left(z,x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆ{W}_{n}^{1,p}\left(Z\right),$

where ${j}_{âˆ’}\left(z,x\right)=j\left(z,{\mathrm{Î½}}_{âˆ’}\left(x\right)\right)$ for all $zâˆˆ\mathbb{R}$, $xâˆˆ\mathbb{R}$ which is locally Lipschitz. Then we have another nontrivial solution ${x}_{âˆ—}âˆˆ{W}_{n}^{1,p}\left(Z\right)$ which is a local minimum of ${\mathrm{Ï†}}_{âˆ’}$ and of Ï† too.â€ƒâ–¡

## 4 Existence of the third nontrivial solution

In this section, we prove the existence of the third solution. Then we give the new assumptions which differ slightly from $A\left(j\right)$(v):

${A}^{â€²}\left(j\right)$:

1. (i)

$zâ†¦j\left(z,x\right)$ is measurable for all $xâˆˆ\mathbb{R}$;

2. (ii)

$xâ†¦j\left(z,x\right)$ is locally Lipschitz for a.e. $zâˆˆZ$;

3. (iii)

$|u|â‰¤\mathrm{Î³}\left(z\right)+C{|x|}^{pâˆ’1}$ for a.e. $zâˆˆZ$, all $xâˆˆ\mathbb{R}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Î³}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ and $C>0$;

4. (iv)

${limâ€‰sup}_{|x|â†’\mathrm{âˆž}}\frac{u}{{|x|}^{pâˆ’2}x}â‰¤\mathrm{Ï‰}\left(z\right)$ for a.e. $zâˆˆZ$, all $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Ï‰}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ satisfying $\mathrm{Ï‰}\left(z\right)â‰¤\mathrm{Î±}$ a.e. in Z and $\mathrm{Ï‰}\left(z\right)<\mathrm{Î±}$ in some set of positive measure;

5. (v)

$\mathrm{Î·}\left(z\right)â‰¤{limâ€‰inf}_{xâ†’0}\frac{u}{{|x|}^{pâˆ’2}x}$ for a.e. $zâˆˆZ$, all $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, with $\mathrm{Î·}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$ satisfying ${\mathrm{Î»}}_{1}+\mathrm{Î±}â‰¤\mathrm{Î·}\left(z\right)$ a.e. in Z and ${\mathrm{Î»}}_{1}+\mathrm{Î±}<\mathrm{Î·}\left(z\right)$ in some set of positive measure, ${\mathrm{Î»}}_{1}$ is the first eigenvalue of $âˆ’{\mathrm{Î”}}_{p}$ with the Robin boundary condition;

6. (vi)

$uxâ‰¥0$ for a.e. $zâˆˆZ$, all $xâˆˆ\mathbb{R}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$.

### Theorem 4.1

Let assumptions${A}^{â€²}\left(j\right)$(i)-(vi) hold. Then we can find three nontrivial solutions${x}_{0}âˆˆint\left({C}_{+}\right)$, ${x}_{âˆ—}âˆˆâˆ’int\left({C}_{+}\right)$, and${y}_{0}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$of (1.1).

### Proof

From Theorem 3.1, we have two constant sign solutions ${x}_{0}âˆˆint\left({C}_{+}\right)$ and ${x}_{âˆ—}âˆˆâˆ’int\left({C}_{+}\right)$ which are the local minimizers of ${\mathrm{Ï†}}_{+}$ and of ${\mathrm{Ï†}}_{âˆ’}$, also of Ï†. We may assume that ${x}_{0}$ is the only nontrivial critical point of ${\mathrm{Ï†}}_{+}$ and ${x}_{âˆ—}$ is the only nontrivial critical point of ${\mathrm{Ï†}}_{âˆ’}$. In fact, if there exists another nontrivial critical point ${x}_{1}$ of ${\mathrm{Ï†}}_{+}$, . Then, by a similar discussion, ${x}_{1}âˆˆint\left({C}_{+}\right)$ and it solves (1.1). Thus we have a third nontrivial solution, a.e. ${y}_{0}={x}_{1}$.

Moreover, as for Ï†, we see that Ï† is coercive and so we can easily prove the Palais-Smale condition. In fact, as in the proof of Theorem 3.1, for a.e. $zâˆˆZ$, all $xâˆˆ\mathbb{R}$, we have

$j\left(z,x\right)<\frac{1}{p}\left(\mathrm{Ï‰}\left(z\right)+\mathrm{Îµ}\right){|x|}^{p}+{\mathrm{Î´}}_{\mathrm{Îµ}}\left(z\right)|x|,$

where Ï‰ satisfies (iii), ${\mathrm{Î´}}_{\mathrm{Îµ}}âˆˆ{L}^{\mathrm{âˆž}}{\left(Z\right)}_{+}$, .

Then, using Lemma 3.5, we have

$\begin{array}{rcl}\mathrm{Ï†}\left(x\right)& â‰¥& \frac{1}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{1}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’\frac{1}{p}{âˆ«}_{Z}\mathrm{Ï‰}\left(z\right){|x\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dzâˆ’\frac{\mathrm{Îµ}}{p}{âˆ¥xâˆ¥}_{p}^{p}âˆ’C{âˆ¥xâˆ¥}_{p}\\ â‰¥& \frac{1}{p}\left({\mathrm{Îµ}}_{0}âˆ’\mathrm{Îµ}\right){âˆ¥xâˆ¥}_{p}^{p}âˆ’C{âˆ¥xâˆ¥}_{p}.\end{array}$

It follows that Ï† is coercive.

We set $S=\mathrm{âˆ‚}{B}_{\mathrm{Î´}}\left({x}_{0}\right)=\left\{xâˆˆ{W}_{n}^{1,p}\left(Z\right):{âˆ¥xâˆ’{x}_{0}âˆ¥}_{{W}_{n}^{1,p}\left(Z\right)}=\mathrm{Î´}\right\}$, $Q=\left[{x}_{âˆ—},{x}_{0}\right]$, with relative boundary $\mathrm{âˆ‚}Q=\left\{{x}_{âˆ—},{x}_{0}\right\}$. If we choose $0<\mathrm{Î´}<{âˆ¥{x}_{âˆ—}âˆ’{x}_{0}âˆ¥}_{{W}_{n}^{1,p}\left(Z\right)}$. Then S and âˆ‚Q link. In fact, $Sâˆ©\mathrm{âˆ‚}Q=\mathrm{âˆ\dots }$, and for any map $hâˆˆ{C}^{0}\left(Q,{W}_{n}^{1,p}\left(Z\right)\right)$ such that $h{|}_{\mathrm{âˆ‚}Q}=Id$, we can choose some $tâˆˆ\left(0,1\right)$ satisfying

${âˆ¥h\left(t{x}_{âˆ—}+\left(1âˆ’t\right){x}_{0}\right)âˆ’{x}_{0}âˆ¥}_{{W}_{n}^{1,p}\left(Z\right)}=\mathrm{Î´},$

so , S and âˆ‚Q link.

When we choose Î´, we can also assume Î´ satisfy ${inf}_{xâˆˆS}\mathrm{Ï†}>\mathrm{Ï†}\left({x}_{0}\right)$ and ${inf}_{xâˆˆS}\mathrm{Ï†}>\mathrm{Ï†}\left({x}_{âˆ—}\right)$ (${x}_{0}$, ${x}_{âˆ—}$ are local minimizers of Ï†), we may assume that $\mathrm{Ï†}\left({x}_{âˆ—}\right)<\mathrm{Ï†}\left({x}_{0}\right)$. Therefore, we can apply Proposition 2.3; let $\mathrm{Î“}=\left\{hâˆˆ{C}^{0}\left(Q,{W}_{n}^{1,p}\left(Z\right)\right):h{|}_{\mathrm{âˆ‚}Q}=Id\right\}$, produce ${y}_{0}âˆˆ{W}_{n}^{1,p}\left(Z\right)$, a critical point of Ï†, such that

$\begin{array}{c}0âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left({y}_{0}\right),\hfill \\ \mathrm{Ï†}\left({x}_{âˆ—}\right)<\mathrm{Ï†}\left({x}_{0}\right)<\underset{xâˆˆS}{inf}\mathrm{Ï†}â‰¤\mathrm{Ï†}\left({y}_{0}\right)=\underset{hâˆˆ\mathrm{Î“}}{inf}\underset{xâˆˆQ}{sup}\mathrm{Ï†}\left(h\left(x\right)\right).\hfill \end{array}$

From the above inequality, we have , .

From $0âˆˆ\mathrm{âˆ‚}\mathrm{Ï†}\left({y}_{0}\right)$, we know that

$\left\{\begin{array}{l}âˆ’{\mathrm{Î”}}_{p}{y}_{0}\left(z\right)+\mathrm{Î±}{|{y}_{0}\left(z\right)|}^{pâˆ’2}{y}_{0}\left(z\right)âˆˆ\mathrm{âˆ‚}j\left(z,{y}_{0}\left(z\right)\right),\phantom{\rule{1em}{0ex}}zâˆˆZ,\\ {|\mathrm{âˆ‡}{y}_{0}\left(z\right)|}^{pâˆ’2}\frac{\mathrm{âˆ‚}{y}_{0}\left(z\right)}{\mathrm{âˆ‚}n}+b\left(z\right){|{y}_{0}\left(z\right)|}^{pâˆ’2}{y}_{0}\left(z\right)=0,\phantom{\rule{1em}{0ex}}zâˆˆ\mathrm{âˆ‚}Z,\end{array}$
(4.1)

and from the regularity theory (see [24]), we have ${y}_{0}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$, hence (4.1) holds in all $zâˆˆZ$, we get ${y}_{0}âˆˆ{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$.

Finally, we prove that . It is equivalent to proving that there is a path $hâˆˆ\mathrm{Î“}$ such that for all $xâˆˆQ$,

$\mathrm{Ï†}\left(h\left(x\right)\right)<0=\mathrm{Ï†}\left(0\right).$

From Proposition 2.1, recall that $S={W}_{n}^{1,p}\left(Z\right)âˆ©\mathrm{âˆ‚}{B}_{1}$, $\mathrm{âˆ‚}{B}_{1}=\left\{xâˆˆ{L}^{p}\left(Z\right):{âˆ¥xâˆ¥}_{p}=1\right\}$ endowed with the ${W}_{n}^{1,p}\left(Z\right)$-topology. Furthermore, set ${S}_{c}=Sâˆ©{C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$ equipped with the ${C}_{n}^{1}\left(\stackrel{Â¯}{Z}\right)$-topology. Then we can find ${h}_{0}âˆˆ{S}_{c}$ by virtue of the density of ${S}_{c}$ in S in the ${W}_{n}^{1,p}\left(Z\right)$-topology, so $C\left(Q,{S}_{c}\right)$ is dense in $C\left(Q,S\right)$, and

$max\left\{{âˆ¥Dxâˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}ds,xâˆˆ{h}_{0}\left(Q\right)\right\}â‰¤{\mathrm{Î»}}_{1}+\mathrm{Î´}.$
(4.2)

From assumption ${A}^{â€²}\left(j\right)$(v), we can find ${\mathrm{Î´}}_{0}>0$, such that for a.e. $zâˆˆZ$, all $0<|x|<{\mathrm{Î´}}_{0}$, $uâˆˆ\mathrm{âˆ‚}j\left(z,x\right)$, we get

$\mathrm{Î·}\left(z\right)â‰¤\frac{u}{{|x|}^{pâˆ’2}x}.$

So for a.e. $zâˆˆZ$, all $0<|x|<{\mathrm{Î´}}_{0}$,

$\frac{\mathrm{Î·}}{p}{|x|}^{p}
(4.3)

Since ${h}_{0}\left(Q\right)âˆˆ{S}_{c}$, for the ${\mathrm{Î´}}_{0}$, we can find $\mathrm{Îµ}>0$, such that for a.e. $zâˆˆ\stackrel{Â¯}{Z}$, $xâˆˆ{h}_{0}\left(Q\right)$, we have

$\mathrm{Îµ}|x\left(z\right)|â‰¤{\mathrm{Î´}}_{0}.$
(4.4)

Then let $\mathrm{Î´}>0$ be such that ${\mathrm{Î»}}_{1}+\mathrm{Î±}+\mathrm{Î´}<\mathrm{Î·}$, from (4.2), (4.3), (4.4), and ${âˆ¥xâˆ¥}_{p}=1$, we have

$\begin{array}{rcl}\mathrm{Ï†}\left(\mathrm{Îµ}x\right)& =& \frac{{\mathrm{Îµ}}^{p}}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}{\mathrm{Îµ}}^{p}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{{\mathrm{Îµ}}^{p}}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’{âˆ«}_{Z}j\left(z,\mathrm{Îµ}x\left(z\right)\right)\phantom{\rule{0.2em}{0ex}}dz\\ â‰¤& \frac{{\mathrm{Îµ}}^{p}}{p}{âˆ¥Dxâˆ¥}_{p}^{p}+\frac{\mathrm{Î±}{\mathrm{Îµ}}^{p}}{p}{âˆ¥xâˆ¥}_{p}^{p}+\frac{{\mathrm{Îµ}}^{p}}{p}{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}dsâˆ’\frac{\mathrm{Î·}{\mathrm{Îµ}}^{p}}{p}{âˆ¥xâˆ¥}_{p}^{p}\\ =& \frac{{\mathrm{Îµ}}^{p}}{p}\left({âˆ¥Dxâˆ¥}_{p}^{p}+{âˆ«}_{\mathrm{âˆ‚}Z}b\left(z\right){|x|}^{p}\phantom{\rule{0.2em}{0ex}}ds\right)+\frac{\mathrm{Î±}âˆ’\mathrm{Î·}}{p}{\mathrm{Îµ}}^{p}\\ â‰¤& \frac{{\mathrm{Î»}}_{1}+\mathrm{Î´}+\mathrm{Î±}âˆ’\mathrm{Î·}}{p}{\mathrm{Îµ}}^{p}<0.\end{array}$

We consider the continuous path ${h}_{\mathrm{Îµ}}=\mathrm{Îµ}{h}_{0}$, then for all $xâˆˆQ$,

$\mathrm{Ï†}\left({h}_{\mathrm{Îµ}}\left(x\right)\right)<0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆQ.$

Next recall that Ï† is coercive and satisfies the Palais-Smale condition. From the discussion, we set $a=\mathrm{Ï†}\left({x}_{0}\right)=inf\mathrm{Ï†}<0$, $b=0$, Ï† has no critical points in ${\mathrm{Ï†}}^{âˆ’1}\left(a,b\right)$, ${K}_{a}=\left\{{x}_{0}\right\}$. Then with the help of Proposition 2.5, there exists a deformation $\mathrm{Î¦}:\left[0,1\right]Ã—{\mathrm{Ï†}}^{b}â†’{\mathrm{Ï†}}^{b}$ such that

$\begin{array}{r}\mathrm{Î¦}\left(t,â‹\dots \right){|}_{{K}_{a}}=Id,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}tâˆˆ\left[0,1\right],\\ \mathrm{Î¦}\left(1,{\mathrm{Ï†}}^{b}\right)âŠ†{\mathrm{Ï†}}^{a}âˆª{K}_{a},\\ \mathrm{Ï†}\left(\mathrm{Î¦}\left(t,x\right)\right)â‰¤\mathrm{Ï†}\left(x\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\left(t,x\right)âˆˆ\left[0,1\right]Ã—{\mathrm{Ï†}}^{b}.\end{array}$
(4.5)

In fact, the continuous path Î“ can be seen as $\mathrm{Î“}=\left\{hâˆˆ{C}^{0}\left(\left[0,1\right],{W}_{n}^{1,p}\left(Z\right)\right):h\left(0\right)={x}_{âˆ—},h\left(1\right)={x}_{0}\right\}$. Then we define ${h}_{1}:\left[0,1\right]â†’{\mathrm{Ï†}}^{b}$ by

${h}_{1}\left(t\right)=\mathrm{Î¦}\left(t,\mathrm{Îµ}{u}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}tâˆˆ\left[0,1\right].$

Then it is a continuous path, so from (4.5), we have

$\begin{array}{c}{h}_{1}\left(0\right)=\mathrm{Î¦}\left(0,\mathrm{Îµ}{u}_{1}\right)=\mathrm{Îµ}{u}_{1},\hfill \\ {h}_{1}\left(1\right)=\mathrm{Î¦}\left(1,\mathrm{Îµ}{u}_{1}\right)={x}_{0}\phantom{\rule{1em}{0ex}}\left({\mathrm{Ï†}}^{a}=\mathrm{âˆ\dots },{K}_{a}=\left\{{x}_{0}\right\}\right),\hfill \\ \mathrm{Ï†}\left({h}_{1}\left(t\right)\right)=\mathrm{Ï†}\left(\mathrm{Î¦}\left(t,\mathrm{Îµ}{u}_{1}\right)\right)â‰¤\mathrm{Ï†}\left(\mathrm{Îµ}{u}_{1}\right)<0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}tâˆˆ\left[0,1\right]\phantom{\rule{0.25em}{0ex}}\left(\mathrm{Ï†}{|}_{{h}_{\mathrm{Îµ}}}<0\right).\hfill \end{array}$

Thus, we construct a continuous path ${h}_{1}$ joining $\mathrm{Îµ}{u}_{1}$ and ${x}_{0}$ such that

$\mathrm{Ï†}{|}_{{h}_{1}}<0.$

Similarly, we construct a continuous path ${h}_{2}$ joining $âˆ’\mathrm{Îµ}{u}_{1}$ and ${x}_{âˆ—}$ such that

$\mathrm{Ï†}{|}_{{h}_{2}}<0.$

Then we join ${h}_{2}$, ${h}_{\mathrm{Îµ}}$, ${h}_{1}$, and we construct a continuous path $hâˆˆ\mathrm{Î“}$ such that

$\mathrm{Ï†}{|}_{h}<0.$

It follows that $\mathrm{Ï†}\left({y}_{0}\right)<0=\mathrm{Ï†}\left(0\right)$ and so .

Therefore, we find the third nontrivial solution of (1.1).â€ƒâ–¡

## 5 Open related questions

Consider the problem

$\left\{\begin{array}{l}âˆ’{\mathrm{Î”}}_{p}x+\mathrm{Î±}\left(z\right){|x|}^{pâˆ’2}xâˆˆ\mathrm{âˆ‚}j\left(z,x\right),\phantom{\rule{1em}{0ex}}zâˆˆZ,\\ {|\mathrm{âˆ‡}x|}^{pâˆ’2}\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}+b\left(z\right){|x|}^{pâˆ’2}x=0,\phantom{\rule{1em}{0ex}}zâˆˆ\mathrm{âˆ‚}Z,\end{array}$
(5.1)

where $ZâŠ‚{\mathbb{R}}^{N}$ is a bounded domain with ${C}^{2}$-boundary âˆ‚Z, ${\mathrm{Î”}}_{p}x=\text{div}\left({|\mathrm{âˆ‡}x|}^{pâˆ’2}\mathrm{âˆ‡}x\right)$ ($1) is the p-Laplacian operator, $\mathrm{Î±}\left(z\right),b\left(z\right)âˆˆ{L}^{\mathrm{âˆž}}\left(\mathrm{âˆ‚}Z\right)$, $b\left(z\right)â‰¥0$, and on âˆ‚Z. $j\left(z,x\right)$ is a measurable potential function on $ZÃ—\mathbb{R}$, which is locally Lipschitz in the $xâˆˆ\mathbb{R}$, $\mathrm{âˆ‚}j\left(z,x\right)$ stands for the generalized subdifferential of $xâ†¦j\left(z,x\right)$. Also $\frac{\mathrm{âˆ‚}x}{\mathrm{âˆ‚}n}$ denotes the outer normal derivative of x with respect to âˆ‚Z.

Whether problem (5.1) has more solutions and whether it has oscillating solutions, we will discuss in the future.

## Authorâ€™s contributions

The author wrote, read, and approved the final manuscript.

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## Acknowledgements

The author was supported by Tianyuan Foundation for Mathematics under Grant No. 11326098 of National Natural Science Foundation and Doctor Scientific Research Startup Foundation of Harbin Normal University No. XKB 201311.

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Zhang, J. Existence of solutions for nonlinear Robin problems with the p-Laplacian and hemivariational inequality. Bound Value Probl 2014, 257 (2014). https://doi.org/10.1186/s13661-014-0257-5