Three Weak Solutions for a Neumann p(x)-Laplacian-like Problem with Two Control Parameters

Abstract

We establish the existence of at least three weak solutions for a Neumann $p\left(x\right)$-Laplacian-like problem with two control parameters. Our main result is due to the critical theorem of Bonanno and Marano.

1. Introduction

Research on the $p\left(x\right)$-Laplacian operator is very active in many areas of physics and applied mathematics. This interest stems from the fact that this operator is involved, for example, in electrorheological fluids (Ružička [1]), elastic mechanics (Zhikov [2]), stationary thermorheological viscous flows of non-Newtonian fluids (Rajagopal-Ružička [3]), image processing (Chen-Levine-Rao [4]), and mathematical description of the processes filtration of barotropic gas through a porous medium (Antontsev-Shmarev [5]).

We mention that capillary action is succinctly described by interaction phenomena that occur at the interface of two immiscible liquids, either liquid–air or liquid–surface. This is caused by the surface tension between the different phases present. Recently, attention has been focused on the study of capillarity. In addition to the interest in natural phenomena such as the motion of water droplets, bubbles, and waves, this growing interest is also fueled by the importance of these phenomena in various practical fields such as industry, biomedical, pharmaceutical, and microfluidic systems. A lot of research has been done to study capillary action. For example, in [6], Rodrigues used the Mountain pass lemma and the Fountain theorem to investigate the existence of non-trivial solutions to the following problem

$$\left\{\begin{array}{c}-div\left(\left(1+\frac{{|\nabla u|}^{p\left(x\right)}}{\sqrt{1+{|\nabla u|}^{2p\left(x\right)}}}\right){|\nabla u|}^{p\left(x\right)-2}\nabla u\right)=\lambda f(x,u),x\in \Omega ,\hfill \\ u=0,x\in \partial \Omega ,\hfill \end{array}\right.$$

where $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ is a bounded regular domain, $\lambda$ is a positive parameter, and f is a Carathéodory function.

Moreover, S. Shokooh, G.A. Afrouzi, and S. Heidarkhani in [7] established the existence of three weak solutions of the following problem

$$\left\{\begin{array}{c}-div\left(\left(1+\frac{{|\nabla u|}^{p\left(x\right)}}{\sqrt{1+{|\nabla u|}^{2p\left(x\right)}}}\right){|\nabla u|}^{p\left(x\right)-2}\nabla u\right)+a\left(x\right){\left|u\right|}^{p\left(x\right)-2}u=\lambda f(x,u)+\mu g(x,u)\mathrm{in}\Omega ,\hfill \\ \frac{\partial u}{\partial \nu }=0\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ is a bounded domain with $C^1$-class boundary of class, $\nu$ is the outer unit normal to $\partial \Omega ,\lambda >0,\mu \ge 0,a\in L^{\infty }(\Omega )$ with $essin{f}_{x\in \Omega }a\ge 0$, $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are $L^1$-Carathéodory functions, and $p\in C\left(\overline{\Omega }\right)$ satisfies the condition

$$N<p^{-}:=\underset{x\in \overline{\Omega }}{inf}p\left(x\right)\le p^{+}:=\underset{x\in \overline{\Omega }}{sup}p\left(x\right)<+\infty .$$

For more details about these kind of operators, the reader can be referred to Zhou-Ge [8], Ge [9], and Heidarkhani-Afrouzi-Moradi [10].

The aim of this paper is to show the existence of three weak solutions for the following problem originated from a capillary phenomena

$$\left\{\begin{array}{cc}\tilde{L}\left(w\right)={\alpha f\left(x\right)\left|w\right|}^{q\left(x\right)-2}w+\mu g\left(x\right){\left|w\right|}^{l\left(x\right)-2}w\hfill & \mathrm{in}\mathrm{Q},\hfill \\ \\ \frac{\partial w}{\partial \nu }=0\hfill & \mathrm{on}\partial \mathrm{Q},\hfill \end{array}\right.$$

(1)

where $\tilde{L}\left(w\right)=-\mathrm{div}\left((1+\frac{{|\nabla w|}^{p\left(x\right)}}{\sqrt{1+{|\nabla w|}^{2p\left(x\right)}}}\right){|\nabla w|}^{p\left(x\right)-2}w)+b\left(x\right){\left|w\right|}^{p\left(x\right)-2}w$, $\mathrm{Q}$ is a bounded domain in ${\mathbb{R}}^N$ ($N\ge 2$) with smooth boundary $\partial \mathrm{Q}$, $\nu$ is the outer unit normal to $\partial \Omega$, $\alpha >0$ and $\mu \ge 0$ are two parameters, $f,g$ are two indefinite weights in a generalized Lebesgue space $L^{s_1\left(x\right)}\left(\mathrm{Q}\right)$ and $L^{s_2\left(x\right)}\left(\mathrm{Q}\right)$, and the functions $p,q,s\in C\left(\overline{\mathrm{Q}}\right)$ satisfy some suitable assumptions that we will mention later. Note that the cases of a source term containing an indefinite weight for $p^{+}<N$ has never been treated in the literature.

In order to introduce our result, we set

$$C_{+}\left(\overline{\mathrm{Q}}\right):=\{h\mid h\in C\left(\overline{\mathrm{Q}}\right),h\left(x\right)>1,forallx\in \overline{\mathrm{Q}}\}.$$

For $\eta >0$ and $h\in C_{+}\left(\overline{\mathrm{Q}}\right)$, let

$$h^{-}:=\underset{x\in \mathrm{Q}}{inf}h\left(x\right),h^{+}:=\underset{x\in \mathrm{Q}}{sup}h\left(x\right)$$

and

$${\left[\eta \right]}^h:=max\{{\eta }^{h^{-}},{\eta }^{h^{+}}\},{\left[\eta \right]}_h:=min\{{\eta }^{h^{-}},{\eta }^{h^{+}}\}.$$

Therefore, ${\left[\eta \right]}^{\frac{1}{h}}:=max\{{\eta }^{\frac{1}{h^{+}}},{\eta }^{\frac{1}{h^{-}}}\}$ and ${\left[\eta \right]}_{\frac{1}{h}}:=min\{{\eta }^{\frac{1}{h^{+}}},{\eta }^{\frac{1}{h^{-}}}\}$.

We denote by

$$\delta \left(x\right):=sup\{\delta >0\mid B(x,\delta )\subseteq \mathrm{Q}\}\mathrm{for} \mathrm{all}x\in \mathrm{Q}.$$

Here, B is the ball centered at x and of radius $\delta$. Therefore, there exists $x_0\in \mathrm{Q}$ verifying $B(x_0,D)\subseteq \mathrm{Q}$, with ${\displaystyle D=\underset{x\in \mathrm{Q}}{sup}\delta \left(x\right).}$

The following hypotheses will be necessary for this paper:

(A)

$max(q^{+},l^{+})<p^{-}\le p^{+}<N<min(s_1\left(x\right),s_2\left(x\right))$  $\mathrm{for}\mathrm{all}x\in \mathrm{Q}.$

(A₁)

Assume that $f\in L^{s\left(x\right)}\left(\mathrm{Q}\right)$ satisfies the following

$$f\left(x\right):=\left\{\begin{array}{c}\le 0,\mathrm{for}x\in \mathrm{Q}\setminus B(x_0,D),\hfill \\ \ge f_0,\mathrm{for}x\in B(x_0,\frac{D}{2}),\hfill \\ >0,\mathrm{for}x\in B(x_0,D)\setminus B(x_0,\frac{D}{2}),\hfill \end{array}\right.$$

and $g\in L^{s_2\left(x\right)}(\Omega )$ satisfies the following

$$g\left(x\right):=\left\{\begin{array}{c}\le 0,\mathrm{for}x\in \Omega \setminus B(x_0,D),\hfill \\ \ge g_0,\mathrm{for}x\in B(x_0,\frac{D}{2}),\hfill \\ >0,\mathrm{for}x\in B(x_0,D)\setminus B(x_0,\frac{D}{2}),\hfill \end{array}\right.$$

where $f_0$ and $g_0$ are positive constants.

In what follows, let

$${\displaystyle L:=m(D^N-{\left(\frac{D}{2}\right)}^N),m:=\frac{{\pi }^{\frac{N}{2}}}{\frac{N}{2}\mathsf{\Gamma }\left(\frac{N}{2}\right)}\mathrm{and}M:=max(\frac{2}{p^{-}},\frac{{\parallel b\parallel }_{\infty }}{p^{-}}),}$$

where $\mathsf{\Gamma }$ denotes the Euler function. Moreover, let $c_1,c_2>0$ be the best constants for which inequalities (5) below hold. The main result of our work is the following.

Theorem 1. 

Suppose that assertions $\left(\mathbf{A}\right)$ and $\left({\mathbf{A}}_{\mathbf{1}}\right)$ hold, then there exist $r>0$ and $d>0$ such that

$$r<\frac{1}{p^{+}}{\left[\frac{2d}{D}\right]}_{p}L$$

(2)

and

$$\begin{array}{ccc}\hfill {\overline{w}}_r& :=& \frac{1}{r}\left\{\frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q+\frac{\mu {\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}}{\alpha l^{-}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l\right\}\hfill \\ & <& {\gamma }_d:=\frac{\frac{1}{q^{+}}{f}_0{\left[d\right]}_{q}m{\left(\frac{D}{2}\right)}^N}{M\left[{\left[\frac{2d}{D}\right]}^{p}L+{\left[d\right]}^{p}L+{\left[d\right]}^{p}m{\left(\frac{D}{2}\right)}^N+\left|\mathrm{Q}\right|\right]}.\hfill \end{array}$$

(3)

Then, for every ${\displaystyle \alpha \in {\overline{\mathsf{\Lambda }}}_r:=\left(\frac{1}{{\gamma }_d},\frac{1}{{\overline{w}}_r}\right),}$ when $\mu \in \left[0,\frac{l^{-}-\alpha \frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q}{{\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l}\right),$ problem (1) admits at least three weak solutions.

Remark 1. 

When $r=1,$ then conditions of Theorem 1 become as follows: there exists $d>0$ such that

$$p^{+}<\left[\frac{2d}{D}\right]L$$

and

$$\begin{array}{c}\hfill {\overline{w}}_1=\left\{\frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}+\frac{\mu {\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}}{\alpha l^{-}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}\right\}<{\gamma }_d.\end{array}$$

Then, for every ${\displaystyle \alpha \in \overline{\mathsf{\Lambda }}:=\left(\frac{1}{{\gamma }_d},\frac{1}{{\overline{w}}_r}\right),}$ when $\mu \in [0,\frac{l^{-}-\alpha \frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}}{{\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}})$, problem (1) admits at least three weak solutions.

This work is structured as follows: we provide some background results and preliminary results on the Sobolev spaces with variable exponents in Section 2, while the proof of our major result is covered in Section 3. Section 4 is devoted to examples.

2. Background Set Up

We review a few characteristics and definitions of variable exponent Sobolev spaces in this section. We refer the reader to Fan-Zhao [11], Rădulescu [12], and Rădulescu-Repovš [13] for a deeper treatment of these spaces and to Papageorgiou-Rădulescu-Repovš [14] for the remaining background material.

Let $p\in C_{+}\left(\overline{\mathrm{Q}}\right)$ be such that

$${\displaystyle 1<p^{-}:=\underset{x\in \overline{\mathrm{Q}}}{min}p\left(x\right)\le p^{+}:=\underset{x\in \overline{\mathrm{Q}}}{max}p\left(x\right)<+\infty .}$$

The Lebesgue space with variable exponent is defined as follows

$$L^{p\left(x\right)}\left(\mathrm{Q}\right):=\{w\mid w:\mathrm{Q}\to \mathbb{R}\mathrm{measurable},{\int }_{\mathrm{Q}}{\left|w\left(x\right)\right|}^{p\left(x\right)}dx<\infty \}.$$

We equip the above space with the Luxemburg norm

$${\left|w\right|}_{p\left(x\right)}:=inf\left\{\eta >0\mid {\int }_{\mathrm{Q}}|\frac{w\left(x\right)}{\eta }{|}^{p\left(x\right)}dx\le 1\right\}.$$

Variable exponent Lebesgue spaces are similar to classical Lebesgue spaces in many respects: they are Banach spaces and are reflexive if and only if $1<p^{-}\le q^{+}<\infty$. Moreover, the inclusion between Lebesgue spaces is generalized naturally: if $q_1,q_2$ are such that $p_1\left(x\right)\le p_2\left(x\right),$ a.e. $x\in \mathrm{Q}$, then there exists a continuous embedding

$$L^{p_2}\left(x\right)\left(\mathrm{Q}\right)\hookrightarrow L^{p_1}\left(x\right)\left(\mathrm{Q}\right).$$

For $w\in L^{p\left(x\right)}\left(\mathrm{Q}\right)$ and $v\in L^{p^{\prime }\left(x\right)}\left(\mathrm{Q}\right)$, the Hölder inequality holds

$$\left|{\int }_{\Omega }wvdx\right|\le \left(\frac{1}{p^{-}}+\frac{1}{{\left(p^{\prime }\right)}^{-}}\right){\left|w\right|}_{p\left(x\right)}{\left|v\right|}_{p^{\prime }\left(x\right)},$$

(4)

where ${\displaystyle \frac{1}{p\left(x\right)}+\frac{1}{p^{\prime }\left(x\right)}=1}$.

The modular on the space $L^{p\left(x\right)}\left(\mathrm{Q}\right)$ is the map ${\rho }_{p\left(x\right)}:L^{p\left(x\right)}\left(\mathrm{Q}\right)\to \mathbb{R}$ defined by

$${\rho }_{p\left(x\right)}\left(w\right):={\int }_{\Omega }{\left|w\right|}^{p\left(x\right)}dx.$$

The Sobolev space with non-standard growth is defined as

$$\begin{array}{c}\hfill W^{1,p\left(x\right)}\left(\mathrm{Q}\right):=\left\{w\in L^{p\left(x\right)}\left(\mathrm{Q}\right):|\nabla w|\in L^{p\left(x\right)}\left(\mathrm{Q}\right)\right\},\end{array}$$

equipped with the norm ${\parallel w\parallel }_{1,p\left(x\right)}:={\left|w\right|}_{p\left(x\right)}+{|\nabla w|}_{p\left(x\right)}$.

For any $w\in W^{1,p\left(x\right)}(\Omega )$, define

$$\parallel w\parallel :=inf\left\{\beta >0:{\int }_{\Omega }\left({\left|\frac{\nabla w\left(x\right)}{\beta }\right|}^{p\left(x\right)}+b\left(x\right){\left|\frac{w\left(x\right)}{\beta }\right|}^{p\left(x\right)}\right)dx\le 1\right\}.$$

Then, it is easy to see that $\parallel w\parallel$ is a norm on $W^{1,p\left(x\right)}(\Omega )$ equivalent to ${\parallel u\parallel }_{1,p\left(x\right)}$. In the following, we will use $\parallel ·\parallel$ instead of ${\parallel ·\parallel }_{1,p\left(x\right)}$ on $X=W^{1,p\left(x\right)}(\Omega )$.

The modular on the space X is the map ${\rho }_{p\left(x\right)}:X\to \mathbb{R}$ defined by

$${\rho }_{p\left(x\right)}\left(w\right):={\int }_{\mathrm{Q}}\left({|\nabla w|}^{p\left(x\right)}+b\left(x\right){\left|w\right|}^{p\left(x\right)}\right)dx.$$

This mapping satisfies some useful properties mentioned below from Fan-Zhao [11].

Proposition 1. 

For every $w,w_n\in X$, the following statements hold:

(1)

$\parallel w\parallel <1(resp.=1,>1)⟺{\rho }_{p\left(x\right)}\left(w\right)<1(resp.=1,>1);$

(1)

${[\parallel w\parallel ]}_p\le {\rho }_{p\left(x\right)}\left(w\right)\le {[\parallel w\parallel ]}^p;$

(3)

$\parallel w_n\parallel \to 0(resp.\to \infty )\iff {\rho }_{p\left(x\right)}\left(w_n\right)\to 0(resp.\to \infty ).$

Proposition 2 

(Edmunds-Rakosnik [15]). Let p and q be measurable functions such that $p\in L^{\infty }\left(\mathrm{Q}\right)$ and $1\le p\left(x\right)q\left(x\right)\le \infty$ for a.e. $x\in \mathrm{Q}$. Let $w\in L^{q\left(x\right)}\left(\mathrm{Q}\right)$, $w\ne 0$. Then

$${\left[\right|w|}_{p\left(x\right)q\left(x\right)}{]}_p\le {\left|\right|w|}^{p\left(x\right)}{|}_{q\left(x\right)}\le {\left[\right|w|}_{p\left(x\right)q\left(x\right)}{]}^p.$$

Let us recall that the definition of the critical Sobolev exponent is:

$${\displaystyle p^{*}\left(x\right):=\frac{Np\left(x\right)}{N-p\left(x\right)},\mathrm{if}p\left(x\right)<Nor{p}^{*}\left(x\right):=+\infty ,ifp\left(x\right)\ge N.}$$

Remark 2 

(Kefi [16]). Express the conjugate exponents of the functions $s_1\left(x\right),s_2\left(x\right)$ as $s_1^{\prime }\left(x\right),s_2^{\prime }\left(x\right)$ and set ${\displaystyle \beta \left(x\right).=\frac{s_1\left(x\right)q\left(x\right)}{s_1\left(x\right)-q\left(x\right)},\eta \left(x\right):=\frac{s_2\left(x\right)l\left(x\right)}{s_2\left(x\right)-l\left(x\right)}.}$ Therefore, one has a compact continuous embedding $X\hookrightarrow L^{s_1^{\prime }\left(x\right)q\left(x\right)}\left(\mathrm{Q}\right),X\hookrightarrow L^{s_2^{\prime }\left(x\right)l\left(x\right)}\left(\mathrm{Q}\right),X\hookrightarrow L^{\beta \left(x\right)}\left(\mathrm{Q}\right)$ and $X\hookrightarrow L^{\eta \left(x\right)}\left(\mathrm{Q}\right)$.

In the sequel, let $c_1,c_2>0$ for the best constants such that

$$\begin{array}{c}\hfill {\left|w\right|}_{s_1^{\prime }\left(x\right)q\left(x\right)}\le c_1{\parallel w\parallel and\left|w\right|}_{s_2^{\prime }\left(x\right)l\left(x\right)}\le c_2\parallel w\parallel .\end{array}$$

(5)

To formulate the variational approach to problem (1) we begin by recalling the notion of a weak solution to our problem.

Definition 1. 

$w\in X\setminus \left\{0\right\}$ is a weak solution of problem (1) if

$\begin{array}{c}{\int }_{\mathrm{Q}}\left({|\nabla w|}^{p\left(x\right)-2}\nabla w+\frac{{|\nabla w|}^{2p\left(x\right)-2}\nabla w}{\sqrt{1+{|\nabla w|}^{2p\left(x\right)}}}\right)\nabla vdx+{\int }_{\mathrm{Q}}b\left(x\right){\left|w\right|}^{p\left(x\right)-2}wvdx-\hfill \\ \alpha {\int }_{\mathrm{Q}}{f\left(x\right)\left|w\right|}^{q\left(x\right)-2}wvdx-\mu {\int }_{\mathrm{Q}}g\left(x\right){\left|w\right|}^{l\left(x\right)-2}wvdx=0,forallv\in X.\hfill \end{array}$

Our main tool will be the following critical theorem from Bonanno-Marano [17], which we restate in a more convenient form.

Theorem 2 

(Bonanno-Marano [17], Theorem 3.6). Let X be a reflexive real Banach space and $\mathsf{\Phi }:X\to \mathbb{R}$ a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X. Let $\mathsf{\Psi }:X\to \mathbb{R}$ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

$$\left(a_0\right)\underset{x\in X}{inf}\mathsf{\Phi }\left(x\right)=\mathsf{\Phi }\left(0\right)=\mathsf{\Psi }\left(0\right)=0.$$

Assume that there exist $r>0$ and $\overline{x}\in X$, with $r<\mathsf{\Phi }\left(\overline{x}\right)$, such that:

$$\left(a_1\right)\frac{{\displaystyle \underset{\mathsf{\Phi }\left(x\right)\le r}{sup}\mathsf{\Psi }\left(x\right)}}{r}<\frac{\mathsf{\Psi }\left(\overline{x}\right)}{\mathsf{\Phi }\left(\overline{x}\right)};$$

$$\left(a_2\right)for each\alpha \in {\mathsf{\Lambda }}_r:=\left(\frac{\mathsf{\Phi }\left(\overline{x}\right)}{\mathsf{\Psi }\left(\overline{x}\right)},\frac{r}{{\displaystyle \underset{\mathsf{\Phi }\left(x\right)\le r}{sup}\mathsf{\Psi }\left(x\right)}}\right),the functional\mathsf{\Phi }-\alpha \mathsf{\Psi }\mathit{is}\mathit{coercive}.$$

Then, for each $\alpha \in {\mathsf{\Lambda }}_r$, the functional $\mathsf{\Phi }-\alpha \mathsf{\Psi }$ has at least three distinct critical points in X.

3. Main Result’s Proof

We establish the proof of Theorem 1, denoted by

$$\mathsf{\Psi }\left(w\right):={\int }_{\mathrm{Q}}\left(\frac{1}{q\left(x\right)}{f\left(x\right)\left|w\right|}^{q\left(x\right)}+\frac{\mu }{\alpha }\frac{1}{l\left(x\right)}g\left(x\right){\left|w\right|}^{l\left(x\right)}\right)dx.$$

Let $I_{\alpha ,\mu }:X\to \mathbb{R},$ defined as

$$I_{\alpha ,\mu }\left(w\right)=\mathsf{\Phi }\left(w\right)-\alpha \mathsf{\Psi }\left(w\right),\forall w\in X,$$

Therefore, $I_{\alpha ,\mu }$ defines the Euler–Lagrange functional corresponding to problem (1). Where

$${\displaystyle \mathsf{\Phi }\left(w\right)={\int }_{\mathrm{Q}}\frac{1}{p\left(x\right)}\left({|\nabla w|}^{p\left(x\right)}+\sqrt{1+{|\nabla w|}^{2p\left(x\right)}}+b\left(x\right){\left|w\right|}^{p\left(x\right)}\right)dx,\forall w\in X,}$$

and it is well known that $\mathsf{\Phi }\in C^1(X,\mathbb{R})$. Moreover,

$${\displaystyle {\mathsf{\Phi }}^{\prime }\left(w\right)\left(v\right)={\int }_{\mathrm{Q}}\left({|\nabla w|}^{p\left(x\right)-2}\nabla w+\frac{{|\nabla w|}^{2p\left(x\right)-2}\nabla w}{\sqrt{1+{|\nabla w|}^{2p\left(x\right)}}}\right)\nabla vdx+{\int }_{\Omega }b\left(x\right){\left|w\left(x\right)\right|}^{p\left(x\right)-2}w\left(x\right)v\left(x\right)dx,\forall w,v\in X.}$$

We recall that $\mathsf{\Phi }$ is convex and sequentially weakly lower semi-continuous and ${\mathsf{\Phi }}^{\prime }:X\to X^{*}$ is a homeomorphism, see Rodrigues [6]. Using Proposition 2, $\mathsf{\Psi }$ is well defined since we have for all $w\in X,$

$$\begin{array}{ccc}\hfill \left|\mathsf{\Psi }\right(w\left)\right|& \le & \frac{1}{q^{-}}{\int }_{\mathrm{Q}}{\left|f\left(x\right)\right|\left|w\right|}^{q\left(x\right)}dx+\frac{\mu }{\alpha l^{-}}{\int }_{\mathrm{Q}}{\left|g\left(x\right)\right|\left|w\right|}^{l\left(x\right)}dx\hfill \\ & \le & \frac{1}{q^{-}}{\left|f\left(x\right)\right|}_{s_1\left(x\right)}{\left|\right|w|}^{q\left(x\right)}{|}_{s_1^{\prime }\left(x\right)}+\frac{\mu }{\alpha l^{-}}{\left|g\left(x\right)\right|}_{s_2\left(x\right)}{\left|\right|w|}^{l\left(x\right)}{|}_{s_2^{\prime }\left(x\right)}\hfill \\ & \le & \frac{1}{q^{-}}{\left|f\left(x\right)\right|}_{s_1\left(x\right)}{\left[\right|w|}_{s_1^{\prime }\left(x\right)q\left(x\right)}{]}^q+\frac{\mu }{\alpha l^{-}}{\left|g\left(x\right)\right|}_{s_2\left(x\right)}{\left|\right[\left|w\right|}_{s_2^{\prime }\left(x\right)l\left(x\right)}{]}^l.\hfill \end{array}$$

In addition, by inequality (5) in Remark 2, one has

$${\displaystyle \left|\mathsf{\Psi }\left(w\right)\right|\le \frac{1}{q^{-}}{\left|f\left(x\right)\right|}_{s\left(x\right)}\left|\right[c_1{\parallel w\parallel ]}^q+\frac{\mu }{\alpha l^{-}}{\left|g\left(x\right)\right|}_{s_2\left(x\right)}\left|\right[c_2{\parallel w\parallel ]}^l,}$$

therefore $\mathsf{\Psi }$ is indeed well defined. Moreover, $\mathsf{\Psi }\left(0\right)=\mathsf{\Phi }\left(0\right)=0$ and ${\mathsf{\Psi }}^{\prime }$ is compact according to Kefi-Irzi-Shomrani (Lemma 3.1 in [18]).

Proof of Theorem 1. 

As we have seen above, the functionals $\mathsf{\Phi }$ and $\mathsf{\Psi }$ satisfy the regularity assumptions of Theorem 2. Now, let $v_d\in X$ be the function defined by

$$v_d:=\left\{\begin{array}{ccc}& 0,& \mathrm{if}x\in \mathrm{Q}\setminus B(x_0,D),\\ & {\displaystyle \frac{2d}{D}(D-|x-x_0\left|\right),}& \mathrm{if}x\in B(x_0,D)\setminus B(x_0,\frac{D}{2}),\\ & d,& \mathrm{if}x\in B(x_0,\frac{D}{2}),\end{array}\right.$$

where $|.|$ denotes the Euclidean norm in ${\mathbb{R}}^N.$

Using inequality $\left(2\right)$ and the assertion on the functions f and g, we have

$$\begin{array}{ccc}\hfill \frac{1}{p^{+}}{\int }_{B(x_0,D)\setminus B(x_0,\frac{D}{2})}|\nabla v_d|dx\le \mathsf{\Phi }\left(v_d\right)& \le & \frac{2}{p^{-}}{\int }_{B(x_0,D)\setminus B(x_0,\frac{D}{2})}|\nabla v_d{|}^{p\left(x\right)}dx+\frac{1}{p^{-}}\left|\mathrm{Q}\right|\hfill \\ & +& {\int }_{B(x_0,D)}b\left(x\right){\left|v_d\right|}^{p\left(x\right)}dx,so\hfill \\ \hfill \frac{1}{p^{+}}{\left[\frac{2d}{D}\right]}_{p}m\left(D^N-{\left(\frac{D}{2}\right)}^N\right)\le \mathsf{\Phi }\left(v_d\right)& \le & \frac{2}{p^{-}}{\left[\frac{2d}{D}\right]}^{p}m\left(D^N-{\left(\frac{D}{2}\right)}^N\right)+\frac{1}{p^{-}}\left|\mathrm{Q}\right|+\frac{{\parallel b\parallel }_{\infty }}{p^{-}}{\left[d\right]}^{p}m{\left(\frac{D}{2}\right)}^N\hfill \\ & +& \frac{{\parallel b\parallel }_{\infty }}{p^{-}}{\left[d\right]}^{p}m\left(D^N-{\left(\frac{D}{2}\right)}^N\right),so\hfill \\ \hfill \frac{1}{p^{+}}{\left[\frac{2d}{D}\right]}_{p}L\le \mathsf{\Phi }\left(v_d\right)& \le & M\left[{\left[\frac{2d}{D}\right]}^{p}L+{\left[d\right]}^{p}L+{\left[d\right]}^{p}m{\left(\frac{D}{2}\right)}^N+\left|\mathrm{Q}\right|\right].\hfill \end{array}$$

Moreover,

$${\displaystyle \mathsf{\Psi }\left(v_d\right)\ge {\int }_{B(x_0,\frac{D}{2})}\left(\frac{f\left(x\right)}{q\left(x\right)}|v_d{|}^{q\left(x\right)}+\frac{\mu g\left(x\right)}{\alpha l\left(x\right)}{\left|v_d\right|}^{l\left(x\right)}\right)dx\ge \frac{1}{q^{+}}{f}_0{\left[d\right]}_{q}m{\left(\frac{D}{2}\right)}^N+\frac{\mu }{\alpha l^{+}}{g}_0{\left[d\right]}_{l}m{\left(\frac{D}{2}\right)}^N}$$

and hence

$$\frac{\mathsf{\Psi }\left(v_d\right)}{\mathsf{\Phi }\left(v_d\right)}>\frac{\frac{1}{q^{+}}{f}_0{\left[d\right]}_{q}m{\left(\frac{D}{2}\right)}^N}{M\left[{\left[\frac{2d}{D}\right]}^{p}L+{\left[d\right]}^{p}L+{\left[d\right]}^{p}m{\left(\frac{D}{2}\right)}^N+\left|\mathrm{Q}\right|\right]}={\gamma }_d.$$

Next, from $r<\frac{1}{p^{+}}{\left[\frac{2d}{D}\right]}_{p}L$, we have $r<\mathsf{\Phi }\left(v_d\right)$. For each $w\in {\mathsf{\Phi }}^{-1}\left((-\infty ,r]\right)$, using assertion $\left(2\right)$ in proposition 1, we obtain

$$\frac{1}{p^{+}}{\left[\parallel w\parallel \right]}_p\le r.$$

(6)

In Proposition 2, inequalities (6) and (5) give

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(w\right)& \le & \frac{1}{q^{-}}{\left|f\right|}_{s_1\left(x\right)}{\left|\right|w|}^{q\left(x\right)}{|}_{s_1^{\prime }\left(x\right)}+\frac{\mu }{\alpha l^{-}}{\left|g\right|}_{s_2\left(x\right)}{\left|\right|w|}^{l\left(x\right)}{|}_{s_2^{\prime }\left(x\right)}\hfill \\ & \le & \frac{1}{q^{-}}{\left|f\right|}_{s_1\left(x\right)}{\left[c_1\parallel w\parallel \right]}^q+\frac{\mu }{\alpha l^{-}}{\left|g\right|}_{s_2\left(x\right)}{\left[c_2\parallel w\parallel \right]}^l\hfill \\ & \le & \frac{1}{q^{-}}{\left|f\right|}_{s_1\left(x\right)}{\left[c_1\right]}^q{\left[{\left(p^{+}\right)}^{\frac{1}{p^{-}}}{\left[r\right]}^{\frac{1}{p}}\right]}^q+\frac{\mu }{\alpha l^{-}}{\left|g\right|}_{s_2\left(x\right)}{\left[c_2\right]}^l{\left[{\left(p^{+}\right)}^{\frac{1}{p^{-}}}{\left[r\right]}^{\frac{1}{p}}\right]}^l\hfill \\ & \le & \frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q+\frac{\mu {\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}}{\alpha l^{-}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l.\hfill \end{array}$$

Therefore,

$${\displaystyle \frac{1}{r}\underset{\mathsf{\Phi }\left(w\right)\le r}{sup}\mathsf{\Psi }\left(w\right)\le {\overline{w}}_r:=\frac{1}{r}\left\{\frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q+\frac{\mu {\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}}{\alpha l^{-}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l\right\}.}$$

Since

$$\mu <\frac{l^{-}-\alpha \frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q}{{\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l},$$

then

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{r}\underset{\mathsf{\Phi }\left(w\right)\le r}{sup}\mathsf{\Psi }\left(w\right)}& \le & \frac{1}{r}\left\{\frac{{\left(p^{+}\right)}^{\frac{q^{+}}{p^{-}}}}{q^{-}}{\left[c_1\right]}^q{\left|f\right|}_{s_1\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^q+\frac{\mu {\left(p^{+}\right)}^{\frac{l^{+}}{p^{-}}}}{\alpha l^{-}}{\left[c_2\right]}^l{\left|g\right|}_{s_2\left(x\right)}{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^l\right\}\hfill \\ & <& \frac{1}{\alpha }.\hfill \end{array}$$

In the following step, we will prove that the energy functional $\mathsf{\Phi }-\alpha \mathsf{\Psi }$ is coercive for all $\alpha >0$. By Remark 2, we have

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(w\right)& \le & \frac{1}{q^{-}}{\int }_{\Omega }{f\left(x\right)\left|w\right|}^{q\left(x\right)}dx+\frac{\mu }{\alpha l^{-}}{\int }_{\Omega }g\left(x\right){\left|w\right|}^{l\left(x\right)}dx\hfill \\ & \le & \frac{1}{q^{-}}{\left|f\right|}_{s_1\left(x\right)}{\left[c_1\parallel w\parallel \right]}^q+\frac{\mu }{\alpha l^{-}}{\left|g\right|}_{s_2\left(x\right)}{\left[c_2\parallel w\parallel \right]}^l.\hfill \end{array}$$

(7)

For $\parallel w\parallel >1,$ assertion $\left(2\right)$ of Proposition 1 and relation (7) give us that

$$\mathsf{\Phi }\left(w\right)-\alpha \mathsf{\Psi }\left(w\right)\ge \frac{1}{p^{+}}{\parallel w\parallel }^{p^{-}}-\mu \frac{1}{q^{-}}{\left|f\right|}_{s_1\left(x\right)}{\left[c_1\parallel w\parallel \right]}^q-\frac{1}{l^{-}}{\left|g\right|}_{s_2\left(x\right)}{\left[c_2\parallel w\parallel \right]}^l.$$

Using the assertion $\left(\mathbf{A}\right)$, we may conclude that $\mathsf{\Phi }\left(w\right)-\alpha \mathsf{\Psi }\left(w\right)$ is coercive. Finally, due to the fact that

$$\overline{\mathsf{\Lambda }}:=\left(\frac{1}{{\gamma }_d},\frac{1}{{\overline{w}}_r}\right)\subseteq \left(\frac{\mathsf{\Phi }\left(v_d\right)}{\mathsf{\Psi }\left(v_d\right)},\frac{r}{{sup}_{\mathsf{\Phi }\left(w\right)\le r}\mathsf{\Psi }\left(w\right)}\right),$$

according to Theorem 2, the functional $\mathsf{\Phi }-\alpha \mathsf{\Psi }$ permits at least three critical points in X that are weak solutions for problem (1) for each $\alpha \in {\overline{\mathsf{\Lambda }}}_r$. The proof of Theorem 1 is now complete. □

4. Examples

Put $\mathrm{Q}=\{(x,y)\in {\mathbb{R}}^2/x^2+y^2<\frac{1}{3}\}$ and let $p(x,y)=x^2+y^2+\frac{3}{2}$. It is obvious that $1<p^{-}=\frac{3}{2}$ and $p^{+}=\frac{11}{6}<2=N$. Assume that $b\equiv 1$, $q\left(x\right)=q$, $l\left(x\right)=l$, $s_1\left(x\right)=s_1$, and $s_2\left(x\right)=s_2$ are a real numbers such that

$$1<max(q,l)<\frac{3}{2}\le \frac{11}{6}<2<min(s_1,s_2).$$

Put

$$\tilde{L}\left(w\right)=-\mathrm{div}((1+\frac{{|\nabla w|}^{x^2+y^2+\frac{3}{2}}}{\sqrt{1+{|\nabla w|}^{2x^2+2y^2+3}}}){|\nabla w|}^{x^2+y^2-\frac{1}{2}}w)+{\left|w\right|}^{x^2+y^2-\frac{1}{2}}w.$$

Problem (1) becomes

$$\left\{\begin{array}{c}\tilde{L}\left(w\right)={\alpha f\left(x\right)\left|w\right|}^{q\left(x\right)-2}w+\mu g\left(x\right){\left|w\right|}^{l\left(x\right)-2}w\mathrm{in}\mathrm{Q},\\ \frac{\partial w}{\partial \nu }=0\mathrm{on}\partial \mathrm{Q},\end{array}\right.$$

and admits three weak distinct solutions.

In this paper, we have shown the existence of three weak distinct solutions to problem (1) in the case where ${sup}_{x\in \overline{\Omega }}p\left(x\right)<N$. An interesting question is a qualitative study of the number of solutions in the case where ${sup}_{x\in \overline{\Omega }}p\left(x\right)\ge N$, which a priori recommends the use of other technics of resolution different from the one introduced in this paper.