A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential

Abstract

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain $\mathcal{D}\subset {\mathbb{R}}^N$ ($N\ge 3$) with homogeneous Dirichlet boundary conditions. A key feature of the system is the presence of a Hardy-type singular term with a variable exponent, where $\delta \left(x\right)$ represents the distance from x to the boundary $\partial \mathcal{D}$. By employing a critical point theorem in the framework of variable exponent Sobolev spaces, we establish the existence of a weak solution whose norm vanishes at zero.

1. Introduction

The study of elliptic partial differential equations involving variable exponent functionals has gained significant attention due to their applications in mathematical physics, elasticity theory, and image processing. In particular, fourth-order elliptic problems have been widely investigated, given their relevance in bending elasticity, nonlinear plate theories, and biophysics.

In this work, we establish the existence of at least one nontrivial weak solution for the fourth-order elliptic problem involving the variable exponent $p\left(x\right)$-biharmonic and $p\left(x\right)$-Laplacian operators:

$$\left\{\begin{array}{cc}{\Delta }_{p\left(x\right)}^{2}w-{\Delta }_{p\left(x\right)}w+\eta \left(x\right){\left|w\right|}^{p\left(x\right)-2}w=\mu \theta \left(x\right){\displaystyle \frac{{\left|w\right|}^{p\left(x\right)-2}w}{\delta {\left(x\right)}^{2p\left(x\right)}}}+\lambda f(x,w),\hfill & \mathrm{in}\mathcal{D},\hfill \\ \\ w=\Delta w=0,\hfill & \mathrm{on}\partial \mathcal{D},\hfill \end{array}\right.$$

(1)

where $\mathcal{D}$ is a bounded domain in ${\mathbb{R}}^N$ ($N\ge 3$) with a smooth boundary $\partial \mathcal{D}$. The function $\eta \left(x\right)\in L^{\infty }\left(\mathcal{D}\right)$ satisfies ${\displaystyle {\eta }_0=\underset{x\in \mathcal{D}}{\mathrm{ess}\inf }\eta \left(x\right)>0}$. The function $p\in C\left(\overline{\mathcal{D}}\right)$ satisfies

$$\underset{x\in \mathcal{D}}{\mathrm{ess}\inf }p\left(x\right):=p^{-}\le p^{+}:=\underset{x\in \mathcal{D}}{\mathrm{ess}\sup }p\left(x\right)<{\displaystyle \frac{N}{2}}.$$

Additionally, $\theta \left(x\right)\in L^{\infty }\left(\mathcal{D}\right)$ with $\underset{x\in \mathcal{D}}{\mathrm{ess}\inf }\theta \left(x\right):={\theta }_0>0$, the parameter $\lambda$ is a positive real value, and $\mu$ is nonnegative one. The function $p\left(x\right)$ belongs to the class:

$$C_{+}\left(\overline{\mathcal{D}}\right):=\left\{z\in C\left(\overline{\mathcal{D}}\right):z^{-}>1\right\}.$$

Moreover, the function $f:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying the growth condition:

$$\left(f\right)\left|f(x,w)\right|\le M_1\left(x\right)+M_2{\left|w\right|}^{s\left(x\right)-1},\mathrm{a}.\mathrm{e}.(x,w)\in \mathcal{D}\times \mathbb{R},$$

where $0<M_1\left(x\right)\in L^{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\left(\mathcal{D}\right)$, $M_2>0$, and $s\left(x\right)\in C_{+}\left(\overline{\mathcal{D}}\right)$ with $s\left(x\right)<p^{*}\left(x\right):={\displaystyle \frac{Np\left(x\right)}{N-2p\left(x\right)}}$.

Our primary goal is to establish the existence of at least one weak solution to problem (1). To achieve this, we employ Ricceri’s variational principle, which provides a powerful framework for dealing with nonlinear and nonhomogeneous differential operators. The application of Ricceri’s theorem, as verified by Bonanno [1], enables us, when $f(x,0)\ne 0$, to demonstrate the existence of solutions under suitable conditions on the parameters $\lambda$ and $\mu$. Next, we establish the existence result for problem (1), in the particular case when $p\left(x\right)=p$, $s\left(x\right)=s$, and f may vanishing in 0. Finally, an example is elaborated on to illustrate our result.

The operators ${\Delta }_{p\left(x\right)}^2$ and ${\Delta }_{p\left(x\right)}$ appearing in (1) represent the $p\left(x\right)$-biharmonic operator and the $p\left(x\right)$ Laplacian, respectively, which generalize the classical p-biharmonic and p-Laplacian operators to the setting of variable exponents. This class of problems has been extensively studied in the context of standard growth conditions, as highlighted in the work of Ferrara and Molica Bisci [2]. Their study investigated the existence of solutions for specific elliptic problems involving the Hardy potential.

$$\left\{\begin{array}{c}-{\Delta }_{p}w=\mu {\displaystyle \frac{{\left|w\right|}^{p-2}w}{{\left|x\right|}^p}}+\lambda g(x,w)\mathrm{in}\mathcal{D},\hfill \\ {\left.w\right|}_{\partial \mathcal{D}}=0,\hfill \end{array}\right.$$

where ${\Delta }_{p}w:=div\left({|\nabla w|}^{p-2}\nabla w\right)$ is the standard p-Laplace operator, $\mathcal{D}$ is a bounded domain in ${\mathbb{R}}^N$ ($N⩾2$) containing the origin and with a smooth boundary $\partial \mathcal{D}$, and $1<p<N$. The function $g:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying the following subcritical growth condition:

$$\left|g(x,t)\right|⩽a_1+a_2{\left|t\right|}^{q-1},\forall (x,t)\in \mathcal{D}\times \mathbb{R},$$

where $a_1,a_2$ are non-negative constants and $q\in ]1,pN/(N-p\left)\right[$. Moreover, $\lambda$ and $\mu$ are two real parameters, with $\lambda >0$ and $\mu \ge 0$. The authors established the existence of a nontrivial solution for both cases where $g(x,0)\ne 0$ and $g(x,0)=0$.

Later, Chaharlang and Razani [3] extended this framework to the biharmonic case by studying the problem:

$$\left\{\begin{array}{cc}{\Delta }_p^{2}w=\mu {\displaystyle \frac{{\left|w\right|}^{p-2}w}{{\left|x\right|}^{2p}}}+\lambda g(x,w)\hfill & \mathrm{in}\mathcal{D},\hfill \\ w=\Delta w=0\hfill & \mathrm{on}\partial \mathcal{D},\hfill \end{array}\right.$$

in this context, ${\Delta }_p^{2}w:=\Delta \left({|\Delta w|}^{p-2}\Delta w\right)$ denotes the p-biharmonic operator, while $\mathcal{D}\subset {\mathbb{R}}^N$ ($N\ge 3$) represents a bounded domain that includes the origin and has a smooth boundary $\partial \mathcal{D}$, with $1<p<N/2$. The function $g:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying the same condition to that of Ferrara–Molica Bisci to ensure the existence of solution. For recent advances on fourth-order problems involving the p-Laplacian, we refer the reader to [4].

A key difference between our work and the literature is the presence of the variable exponent structure in our problem, which introduces additional mathematical challenges. Unlike [2,3], where a standard p-Laplace and p-biharmonic operators are considered, we address a problem involving the $p\left(x\right)$-biharmonic and the $p\left(x\right)$-Laplacian operators with a singular Hardy term. Moreover, our problem extends the results of [2,3] by incorporating a Hardy term with variable exponents. For a deeper understanding of elliptic problems involving multiple solutions for the p-Laplacian operator, we encourage readers to consult the insightful work in [5]. For recent works involving these types of operators, we refer to the works of [6,7,8,9,10,11,12,13].

Throughout the paper and for any $\alpha \in C_{+}\left(\overline{\mathcal{D}}\right)$, we define:

$$\begin{array}{cc}\hfill {\left[c\right]}^{\alpha }& :=max\{c^{{\alpha }^{-}},c^{{\alpha }^{+}}\},\hfill \\ \hfill {\left[c\right]}_{\alpha }& :=min\{c^{{\alpha }^{-}},c^{{\alpha }^{+}}\}.\hfill \end{array}$$

Then, we have the following remark:

Remark 1 

(see [14]).

(1) 

${\left[c\right]}^{{\displaystyle \frac{1}{\alpha }}}=max\left\{c^{{\displaystyle \frac{1}{{\alpha }^{-}}}},c^{{\displaystyle \frac{1}{{\alpha }^{+}}}}\right\},$

(2) 

${\left[c\right]}_{{\displaystyle \frac{1}{\alpha }}}=min\left\{c^{{\displaystyle \frac{1}{{\alpha }^{-}}}},c^{{\displaystyle \frac{1}{{\alpha }^{+}}}}\right\},$

(3) 

${\left[c\right]}_{{\displaystyle \frac{1}{\alpha }}}=b⟺c={\left[b\right]}^{\alpha },{\left[c\right]}^{{\displaystyle \frac{1}{\alpha }}}=b⟺c={\left[b\right]}_{\alpha },$

(4) 

${\left[c\right]}_{\alpha }{\left[\tau \right]}_{\alpha }\le {\left[c\tau \right]}_{\alpha }\le {\left[c\tau \right]}^{\alpha }\le {\left[c\right]}^{\alpha }{\left[\tau \right]}^{\alpha }.$

This document is structured as follows. In the next section, we will introduce preliminary concepts and fundamental results related to Sobolev spaces with variable exponents. The last section is dedicated to proving our main result.

2. Background and Preliminaries

In this section, we recall the definition and key properties of Sobolev spaces with variable exponents. For a more in-depth treatment of these spaces, refer to the works of Fan-Zhao [15] and Edmunds–Rakosnik [16].

The Lebesgue space with a variable exponent is defined as

$$L^{p(·)}\left(\mathcal{D}\right):=\{\omega \mid \omega :\mathcal{D}\to \mathbb{R}\mathrm{is}\mathrm{measurable},\mathrm{and}{\int }_{\mathcal{D}}{\left|\omega \left(x\right)\right|}^{p(·)}dx<\infty \}.$$

This space is equipped with the Luxemburg norm, given by

$${\left|\omega \right|}_{p(·)}:=inf\left\{\sigma >0\mid {\int }_{\mathcal{D}}|{\displaystyle \frac{\omega \left(x\right)}{\sigma }}{|}^{p(·)}dx\le 1\right\}.$$

Variable exponent Lebesgue spaces share several properties with classical Lebesgue spaces. They form Banach spaces and are reflexive if and only if $1<p^{-}\le p^{+}<\infty$. Moreover, these spaces exhibit generalized embedding properties. Specifically, if $p_1(·)\le p_2(·)$ almost everywhere in $\mathcal{D}$, then the continuous embedding

$$L^{p_2(·)}\left(\mathcal{D}\right)\hookrightarrow L^{p_1(·)}\left(\mathcal{D}\right)$$

holds, along with the norm estimate

$${\left|\omega \right|}_{p_1\left(x\right)}\le c_{p_2}{\left|\omega \right|}_{p_2\left(x\right)},$$

for the positive constant $c_{p_2}$.

For functions $\omega \in L^{p(·)}\left(\mathcal{D}\right)$ and ${\omega }_1\in L^{p^0\left(x\right)}\left(\mathcal{D}\right)$, the following Hölder-type inequality is satisfied:

$$\left|{\int }_{\mathcal{D}}\omega {\omega }_{1}dx\right|\le \left({\displaystyle \frac{1}{p^{-}}}+{\displaystyle \frac{1}{{\left(p^0\right)}^{-}}}\right){\left|\omega \right|}_{p(·)}{\left|{\omega }_1\right|}_{p^0(·)},$$

(2)

where the exponents satisfy the relation

$${\displaystyle \frac{1}{p\left(x\right)}}+{\displaystyle \frac{1}{p^0\left(x\right)}}=1.$$

The modular function associated with $L^{p(·)}\left(\mathcal{D}\right)$ is defined as

$${\rho }_{p(·)}\left(\omega \right):={\int }_{\mathcal{D}}{\left|\omega \right|}^{p(·)}dx.$$

Now, consider the Sobolev space with variable exponents, defined by

$$W^{l,p(·)}\left(\mathcal{D}\right):=\left\{\omega \in L^{p(·)}\left(\mathcal{D}\right)\mid D^{\alpha }\omega \in L^{p(·)}\left(\mathcal{D}\right),\left|\alpha \right|\le l\right\},$$

where l is a positive integer, $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N)$ is a multi-index, and

$$\left|\alpha \right|=\sum _{i=1}^N{\alpha }_i,D^{\alpha }\omega ={\displaystyle \frac{{\partial }^{\left|\alpha \right|}\omega }{{\partial }^{{\alpha }_1}{x}_1\cdots {\partial }^{{\alpha }_N}{x}_N}}.$$

The space $W^{l,p(·)}\left(\mathcal{D}\right)$ forms a separable and reflexive Banach space endowed with the norm

$${\parallel \omega \parallel }_{l,p(·)}:=\sum _{\left|\alpha \right|\le l}{\left|D^{\alpha }\omega \right|}_{p(·)}.$$

The subspace $W_0^{l,p(·)}\left(\mathcal{D}\right)$ is defined as the closure of $C_0^{\infty }\left(\mathcal{D}\right)$ in $W^{l,p(·)}\left(\mathcal{D}\right)$. It is known that both $W^{2,p(·)}\left(\mathcal{D}\right)$ and $W_0^{1,p(·)}\left(\mathcal{D}\right)$ are separable and reflexive Banach spaces. We now introduce the functional space defined by

$$X:=W^{2,p(·)}\left(\mathcal{D}\right)\cap W_0^{1,p(·)}\left(\mathcal{D}\right),$$

which is a separable and reflexive Banach space. The associated norm is given by

$$\parallel \omega \parallel =inf\left\{\sigma >0:{\int }_{\mathcal{D}}\left(\eta \left(x\right){\left|{\displaystyle \frac{\omega \left(x\right)}{\sigma }}\right|}^{p(·)}+{\left|{\displaystyle \frac{\nabla \omega \left(x\right)}{\sigma }}\right|}^{p(·)}+{\left|{\displaystyle \frac{\Delta \omega \left(x\right)}{\sigma }}\right|}^{p(·)}\right)dx\le 1\right\}.$$

As established in [17], the norms $\parallel ·\parallel$, ${\parallel ·\parallel }_{2,p(·)}$, and ${|\Delta \omega |}_{p(·)}$ are equivalent on X. Therefore, we work with the normed space $(X,\parallel ·\parallel )$ in the subsequent sections.

The modular function associated with the space X is the mapping ${\rho }_{p(·)}$, which maps elements from X to $\mathbb{R}$ and is defined as

$${\rho }_{p(·)}\left(\omega \right):={\int }_{\mathcal{D}}\left({|\Delta \omega |}^{p(·)}+{|\nabla \omega |}^{p(·)}+\eta \left(x\right){\left|\omega \right|}^{p(·)}\right)dx.$$

It is evident that this mapping satisfies several important properties, some of which are listed below.

Lemma 1. 

For any $\omega ,{\omega }_n\in W^{2,p(·)}\left(\mathcal{D}\right)\cap W_0^{1,p(·)}\left(\mathcal{D}\right)$, the following statements hold:

(1) 

$\parallel \omega \parallel <1(\mathit{resp}.=1,>1)⟺{\rho }_{p(·)}\left(\omega \right)<1(\mathit{resp}.=1,>1),$

(2) 

${[\parallel \omega \parallel ]}_p\le {\rho }_{p(·)}\left(\omega \right)\le {[\parallel \omega \parallel ]}^p,$

(3) 

$\parallel {\omega }_n\parallel \to 0(\mathit{resp}.\to \infty )⟺{\rho }_{p(·)}\left({\omega }_n\right)\to 0(\mathit{resp}.\to \infty ).$

We recall that the critical exponent is defined as follows:

$$p^{*}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{Np\left(x\right)}{N-2p\left(x\right)}},\hfill & \mathrm{if}p\left(x\right)<{\displaystyle \frac{N}{2}},\hfill \\ +\infty ,\hfill & \mathrm{if}p\left(x\right)\ge {\displaystyle \frac{N}{2}}.\hfill \end{array}\right.$$

We now provide the definition of a weak solution to problem (1).

Definition 1. 

A function $w\in X\setminus \left\{0\right\}$ that satisfies $\Delta w=0$ on $\partial \mathcal{D}$ is considered a weak solution to problem (1) if the following holds:

$$\begin{array}{cc}\hfill & {\int }_{\mathcal{D}}{|\Delta w\left(x\right)|}^{p\left(x\right)-2}\Delta w\left(x\right)\Delta v\left(x\right)dx+{\int }_{\mathcal{D}}{|\nabla w\left(x\right)|}^{p\left(x\right)-2}\nabla w\left(x\right)\nabla v\left(x\right)dx\hfill \\ \hfill & +{\int }_{\mathcal{D}}\eta \left(x\right){\left|w\left(x\right)\right|}^{p\left(x\right)-2}w\left(x\right)v\left(x\right)dx-\mu {\int }_{\mathcal{D}}\theta \left(x\right){\displaystyle \frac{{\left|w\left(x\right)\right|}^{p\left(x\right)-2}w\left(x\right)}{\delta {\left(x\right)}^{2p\left(x\right)}}}v\left(x\right)dx\hfill \\ \hfill & =\lambda {\int }_{\mathcal{D}}f(x,w\left(x\right))v\left(x\right)dx.\hfill \end{array}$$

Next, we define

$$\mathcal{V}\left(w\right):={\int }_{\mathcal{D}}F(x,w)dx,\mathrm{where}F(x,w)={\int }_0^{w}f(x,\tau )d\tau .$$

The energy functional associated with problem (1) is given by $I_{\lambda ,\mu }:X\to \mathbb{R}$, where

$$I_{\mu ,\lambda }\left(w\right)={\mathcal{A}}_{\mu }\left(w\right)-\lambda \mathcal{V}\left(w\right),\mathrm{for} \mathrm{all}w\in X$$

(3)

and

$${\mathcal{A}}_{\mu }\left(w\right)={\int }_{\mathcal{D}}{\displaystyle \frac{{|\Delta w\left(x\right)|}^{p\left(x\right)}}{p\left(x\right)}}dx+{\int }_{\mathcal{D}}{\displaystyle \frac{{|\nabla w\left(x\right)|}^{p\left(x\right)}}{p\left(x\right)}}dx+{\int }_{\mathcal{D}}{\displaystyle \frac{{\eta \left(x\right)\left|w\left(x\right)\right|}^{p\left(x\right)}}{p\left(x\right)}}dx-\mu {\int }_{\mathcal{D}}{\displaystyle \frac{\theta \left(x\right)}{p\left(x\right)}}{\displaystyle \frac{{\left|w\left(x\right)\right|}^{p\left(x\right)}}{{\left|\delta \left(x\right)\right|}^{2p\left(x\right)}}}dx.$$

In the reminder, let us recall the $p(·)$-Hardy inequality

Remark 2 

([18]). Assume that $p^{+}<N/2$, then

$${\int }_{\mathcal{D}}{\displaystyle \frac{{|\Delta w\left(x\right)|}^{p\left(x\right)}}{p\left(x\right)}}dx\ge C_H{\int }_{\mathcal{D}}{\displaystyle \frac{{\left|w\left(x\right)\right|}^{p\left(x\right)}}{p\left(x\right)\delta {\left(x\right)}^{2p\left(x\right)}}}dx$$

(4)

for all $w\in W_0^{2,p\left(x\right)}\left(\mathcal{D}\right)$, where $C_H$ is given by

$$C_H={\displaystyle \frac{p^{-}}{p^{+}}}min\left({\left({\displaystyle \frac{N\left(p^{-}-1\right)\left(N-2p^{-}\right)}{{\left(p^{-}\right)}^2}}\right)}^{p^{-}},{\left({\displaystyle \frac{N\left(p^{+}-1\right)\left(N-2p^{+}\right)}{{\left(p^{+}\right)}^2}}\right)}^{p^{+}}\right).$$

The key result we rely on is the following fundamental theorem by Bonanno–Molica Bisci [1], which we state below:

Theorem 1 

([1]). Let X be a reflexive real Banach space, and consider two Gâteaux differentiable functionals $\mathcal{A},\mathcal{V}:X\to \mathbb{R}$. Suppose that $\mathcal{A}$ is strongly continuous, sequentially weakly lower semicontinuous, and coercive, while $\mathcal{V}$ is sequentially weakly upper semicontinuous. For any $r>{inf}_X\mathcal{V}$, define

$$\xi \left(r\right):=\underset{u\in {\mathcal{A}}^{-1}\left(\right]-\infty ,r\left[\right)}{inf}{\displaystyle \frac{\left({sup}_{v\in {\mathcal{A}}^{-1}\left(\right]-\infty ,r\left[\right)}\mathcal{V}\left(v\right)\right)-\mathcal{V}\left(u\right)}{r-\mathcal{A}\left(u\right)}}.$$

Then, for every $r>{inf}_X\mathcal{A}$ and any $\lambda \in (0,1/\xi (r\left)\right)$, the functional $I_{\lambda }:=\mathcal{A}-\lambda \mathcal{V}$, restricted to ${\mathcal{A}}^{-1}\left(\right]-\infty ,r\left[\right)$, attains a global minimum. Moreover, this minimum is a critical point (local minimum) of $I_{\lambda }$ in X.

3. Main Theorem

By using the above Remark, one has:

$${\int }_{\mathcal{D}}{\displaystyle \frac{{\theta \left(x\right)\left|w\left(x\right)\right|}^{p\left(x\right)}}{p\left(x\right)\delta {\left(x\right)}^{2p\left(x\right)}}}dx\le {\displaystyle \frac{{\left|\theta \right|}_{\infty }}{p^{-}{C}_H}}{\int }_{\mathcal{D}}{|\Delta w|}^{p\left(x\right)}dx,$$

combining this with the assumption (2) of Lemma 1, a simple calculation on the expression of ${\mathcal{A}}_{\mu }\left(w\right)$ leads to the following Lemma:

Lemma 2. 

For any $w\in X$, we have

$${\displaystyle \frac{1}{p^{-}}}{[\parallel w\parallel ]}^p\ge {\mathcal{A}}_{\mu }\left(w\right)\ge ({\displaystyle \frac{1}{p^{+}}}-{\displaystyle \frac{{\mu \left|\theta \right|}_{\infty }}{p^{-}{C}_H}}){[\parallel w\parallel ]}_p.$$

It follows that ${\mathcal{A}}_{\mu }$ is well-defined and coercive for

$$\mu \le {\displaystyle \frac{p^{-}{C}_H}{p^{+}{\left|\theta \right|}_{\infty }}}:=K.$$

Furthermore, a straightforward argument confirms that ${\mathcal{A}}_{\mu }$ is a Gâteaux differentiable functional in X that is also weakly lower semicontinuous. Its derivative is given by

$$\begin{array}{cc}\hfill {\mathcal{A}}_{\mu }^{\prime }\left(w\right)\left(v\right)& ={\int }_{\mathcal{D}}{|\Delta w\left(x\right)|}^{p\left(x\right)-2}\Delta w\left(x\right)\Delta v\left(x\right)dx+{\int }_{\mathcal{D}}{|\nabla w\left(x\right)|}^{p\left(x\right)-2}\nabla w\left(x\right)\nabla v\left(x\right)dx\hfill \\ \hfill & +{\int }_{\mathcal{D}}\eta \left(x\right){\left|w\left(x\right)\right|}^{p\left(x\right)-2}w\left(x\right)v\left(x\right)dx-\mu {\int }_{\mathcal{D}}\theta \left(x\right){\displaystyle \frac{{\left|w\left(x\right)\right|}^{p\left(x\right)-2}w\left(x\right)}{\delta {\left(x\right)}^{2p\left(x\right)}}}v\left(x\right)dx.\hfill \end{array}$$

Further, ${inf}_X{\mathcal{A}}_{\mu }={\mathcal{A}}_{\mu }\left(0\right)=\mathcal{V}\left(0\right)=0$ and $\mathcal{V}$ is Gâteaux differentiable, with

$$⟨{\mathcal{V}}^{\prime }\left(w\right),v⟩={\int }_{\mathcal{D}}f(x,w)vdx$$

for all $w,v\in X$. Moreover, we have

Lemma 3. 

The operator ${\mathcal{V}}^{\prime }:X\to X^{*}$ is compact.

Proof. 

By condition $\left(f\right)$ and the compact embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$, where $1<s\left(x\right)<p^{*}\left(x\right)$, we conclude that ${\mathcal{V}}^{\prime }\left(w\right)$ is compact.

Let $\left\{w_n\right\}\subset X$ be a sequence such that $w_n\rightharpoonup w$. Since the embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$ is compact, there is a subsequence of $\left\{w_n\right\}$, still denoted by $\left\{w_n\right\}$, such that $w_n\to w$ strongly in $L^{s\left(x\right)}\left(\mathcal{D}\right)$. Next, we show that the Nemytskii operator $N_f\left(w\right)\left(x\right)=f(x,w\left(x\right))$ is continuous. This holds because $f:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying condition $\left(f\right)$. Thus, $N_f\left(w_n\right)\to N_f\left(w\right)$ in $L^{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\left(\mathcal{D}\right)$. Using Hölder’s inequality (2) and the compact embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$, for all $v\in X$, we have

$$\begin{array}{cc}\hfill |{\mathcal{V}}^{\prime }\left(w_n\right)\left(v\right)-{\mathcal{V}}^{\prime }\left(w\right)\left(v\right)|& =\left|{\int }_{\mathcal{D}}f(x,w_n)vdx-{\int }_{\mathcal{D}}f(x,w)vdx\right|\hfill \\ \hfill & =\left|{\int }_{\mathcal{D}}(f(x,w_n)-f(x,w))vdx\right|\hfill \\ \hfill & \le 2\parallel N_f\left(w_n\right)-N_f\left(w\right){\parallel }_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{\parallel v\parallel }_{s\left(x\right)}\hfill \\ \hfill & \le 2c_s\parallel N_f\left(w_n\right)-N_f\left(w\right){\parallel }_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\parallel v\parallel ,\hfill \end{array}$$

where $c_s$ is the embedding constant for the embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$. Therefore, ${\mathcal{V}}^{\prime }\left(w_n\right)\to {\mathcal{V}}^{\prime }\left(w\right)$ in $X^{*}$, implying that ${\mathcal{V}}^{\prime }$ is completely continuous. Hence, ${\mathcal{V}}^{\prime }$ is compact. □

In what follows, we define

$$\tilde{M}:={\displaystyle \frac{p^{+}{p}^{-}{C}_H}{p^{-}{C}_H-\mu p^{+}{\left|\theta \right|}_{\infty }}}.$$

Our objective is to demonstrate the existence of at least one weak solution to problem (1). The key methodological approach relies on Ricceri’s theorem, as validated by G. Bonanno [1].

Theorem 2. 

Let $f:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ be a Carathéodory function that satisfies condition $\left(f\right)$, with $f(x,0)\ne 0$ in $\mathcal{D}$. Then, for any $\mu \in [0,K[$, where

$$K:={\displaystyle \frac{p^{-}{C}_H}{p^{+}{\left|\theta \right|}_{\infty }}},$$

there is a positive constant

$${\tilde{\lambda }}_{\mu }:=\underset{\gamma >0}{sup}{\displaystyle \frac{{\left[\gamma \right]}_p}{c_s{\left|M_1\left(x\right)\right|}_{\frac{s\left(x\right)}{s\left(x\right)-1}}{\left[\tilde{M}\right]}^{1/p}\gamma +\frac{{\left[c_s\right]}^s{M}_2}{s^{-}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s{\left[\gamma \right]}^s}},$$

such that for any $\lambda \in ]0,{\tilde{\lambda }}_{\mu }[$, problem (1.1) admits at least one nontrivial weak solution ${\tilde{w}}_{\lambda }\in X$, with

$$\underset{\lambda \to 0^{+}}{lim}∥{\tilde{w}}_{\lambda }∥=0.$$

Furthermore, the function

$$h\left(\lambda \right):=I_{\mu ,\lambda }\left({\tilde{w}}_{\lambda }\right),$$

where $I_{\mu ,\lambda }$ is given by (3), is strictly decreasing and remains negative on $]0,{\tilde{\lambda }}_{\mu }[$.

Proof. 

To apply Theorem 1, we begin by fixing $\mu \in [0,K[$ and selecting $\lambda \in ]0,{\tilde{\lambda }}_{\mu }[$. We define the space

$$X:=W^{2,p(·)}\left(\mathcal{D}\right)\cap W_0^{1,p(·)}\left(\mathcal{D}\right),$$

and set $\mathcal{A}:={\mathcal{A}}_{\mu }$. Additionally, let $\mathcal{V}$ denote the functional introduced in Section 2. The proof of the theorem follows these steps:

Step 1. It is well established that the functional $\mathcal{A}:X\to \mathbb{R}$ is continuously Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous for all $\mu \in [0,K[$. Furthermore, it satisfies

$$\underset{w\in X}{inf}\mathcal{A}\left(u\right)=0.$$

In addition, the functional $\mathcal{V}:X\to \mathbb{R}$ is also continuously Gâteaux differentiable. By condition $\left(f\right)$, we have

$$\left|f(x,t)\right|\le M_1\left(x\right)+M_2{\left|t\right|}^{s\left(x\right)-1},\mathrm{a}.\mathrm{e}.(x,t)\in \mathcal{D}\times \mathbb{R}.$$

(5)

As $\lambda$ belongs to $]0,{\tilde{\lambda }}_{\mu }[$, there is a certain $\gamma >0$ such that

$$\lambda <{\tilde{\lambda }}_{\mu }\left(\gamma \right):={\displaystyle \frac{{\left[\gamma \right]}_p}{c_s{\left|M_1\left(x\right)\right|}_{\frac{s\left(x\right)}{s\left(x\right)-1}}{\left[\tilde{M}\right]}^{1/p}\gamma +\frac{{\left[c_s\right]}^s{M}_2}{s^{-}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s{\left[\gamma \right]}^s}}.$$

(6)

From (11), Hölder inequality (2), and the continuous embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$, one has

$$\begin{array}{cc}\hfill \mathcal{V}\left(w\right)={\int }_{\mathcal{D}}F(x,w\left(x\right))\mathrm{d}x& \le {\int }_{\mathcal{D}}(M_1\left(x\right)\left|w\right|+{\displaystyle \frac{M_2}{s\left(x\right)}}{\left|w\right|}^{s\left(x\right)})dx,\hfill \\ & \le |M_1{\left(x\right)|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\parallel w\parallel +{\displaystyle \frac{M_2}{s^{-}}}{\left[{\left|w\right|}_{s\left(x\right)}\right]}^s,\hfill \\ & \le c_s|M_1{\left(x\right)|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\parallel w\parallel +{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[\parallel w\parallel \right]}^s,\hfill \end{array}$$

where $c_s$ is the embedding constant of $X\hookrightarrow L^{s\left(x\right)}\left(\mathcal{D}\right)$. □

Therefore, due to Lemma 2 and Remark 1, one has

$$\parallel w\parallel <{\left[r\tilde{M}\right]}^{1/p}$$

for $w\in X$ such that $\mathcal{A}\left(w\right)<r$. Since

$$\mathcal{V}\left(w\right)\le c_s|M_1{\left(x\right)|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}\parallel w\parallel +{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[\parallel w\parallel \right]}^s,$$

one has

$$\mathcal{V}\left(w\right)<c_s{\left|M_1\left(x\right)\right|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{\left[r\tilde{M}\right]}^{1/p}+{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[{\left[r\tilde{M}\right]}^{1/p}\right]}^s,$$

then

$$\underset{w\in {\mathcal{A}}^{-1}\left(\right]-\infty ,r\left[\right)}{sup}\mathcal{V}\left(w\right)\le c_s{\left|M_1\left(x\right)\right|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{\left[\tilde{M}\right]}^{1/p}{\left[r\right]}^{{\displaystyle \frac{1}{p}}}+{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s[{\left[r\right]}^{{\displaystyle \frac{1}{p}}}{]}^s.$$

Hence,

$${\displaystyle \frac{{sup}_{w\in {\mathcal{A}}^{-1}\left(\right]-\infty ,r\left[\right)}\mathcal{V}\left(w\right)}{r}}\le c_s{\left|M_1\left(x\right)\right|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{\left[\tilde{M}\right]}^{1/p}{\displaystyle \frac{{\left[r\right]}^{\frac{1}{p}}}{r}}+{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s{\displaystyle \frac{{\left[{\left[r\right]}^{\frac{1}{p}}\right]}^s}{r}}.$$

For every $r\in (0,\infty )$, and specifically when selecting $r={\left[\gamma \right]}_p$, we derive

$${\displaystyle \frac{\underset{w\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{sup}\mathcal{V}\left(w\right)}{{\left[\gamma \right]}_p}}\le c_s{\left|M_1\left(x\right)\right|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{\left[\tilde{M}\right]}^{1/p}{\displaystyle \frac{\gamma }{{\left[\gamma \right]}_p}}+{\displaystyle \frac{{\left[c_s\right]}^s{M}_2}{s^{-}}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s{\displaystyle \frac{{\left[\gamma \right]}^s}{{\left[\gamma \right]}_p}}.$$

(7)

Since $0\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)$ and $\mathcal{A}\left(0\right)=\mathcal{V}\left(0\right)=0$, we can define

$$\xi \left({\left[\gamma \right]}_p\right):=\underset{w\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{inf}{\displaystyle \frac{\left(\underset{v\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{sup}\mathcal{V}\left(v\right)\right)-\mathcal{V}\left(w\right)}{{\left[\gamma \right]}_p-\mathcal{A}\left(w\right)}}\le {\displaystyle \frac{\underset{v\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{sup}\mathcal{V}\left(v\right)}{{\left[\gamma \right]}_p}}.$$

Now, using inequalities (6) and (7), we obtain

$$\xi \left({\left[\gamma \right]}_p\right)\le {\displaystyle \frac{\underset{v\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{sup}\mathcal{V}\left(v\right)}{{\left[\gamma \right]}_p}}<{\displaystyle \frac{1}{\lambda }}.$$

This implies

$$\lambda \in \left]0,{\displaystyle \frac{{\left[\gamma \right]}_p}{c_s{\left|M_1\left(x\right)\right|}_{\frac{s\left(x\right)}{s\left(x\right)-1}}{\left[\tilde{M}\right]}^{1/p}\gamma +\frac{{\left[c_s\right]}^s{M}_2}{s^{-}}{\left[{\left[\tilde{M}\right]}^{1/p}\right]}^s{\left[\gamma \right]}^s}}\right[\subseteq \left]0,{\displaystyle \frac{1}{\xi \left({\left[\gamma \right]}_p\right)}}\right[.$$

Applying Theorem 2, we deduce the existence of a critical point $w_{\lambda }\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)$ for $I_{\mu ,\lambda }$, which serves as a global minimizer of $I_{\mu ,\lambda }$ within ${\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)$. Moreover, since $f(x,0)\ne 0$ in $\mathcal{D}$, it follows that $w_{\lambda }\ne 0$.

Step 2. We now demonstrate that $\underset{\lambda \to 0^{+}}{lim}\parallel w_{\lambda }\parallel =0$ and that the function

$$h\left(\lambda \right):=I_{\mu ,\lambda }\left(w_{\lambda }\right)$$

is strictly decreasing and negative in $]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[$. Since the functional $\mathcal{A}$ is coercive, there is a constant $C>0$ such that

$$\parallel w_{\lambda }\parallel \le C,$$

for all $w_{\lambda }\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)$ and any $\lambda \in ]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[$.

Furthermore, due to the compactness of ${\mathcal{V}}^{\prime }$, there is a constant $k>0$ such that

$$\left|{\mathcal{V}}^{\prime }\left(w_{\lambda }\right)\left(w_{\lambda }\right)\right|\le \parallel {\mathcal{V}}^{\prime }\left(w_{\lambda }\right)\parallel \parallel w_{\lambda }\parallel <k{C}^2,$$

(8)

for every $\lambda \in ]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[$.

Furthermore, since $w_{\lambda }$ is a critical point of $I_{\mu ,\lambda }$, we obtain

$$I_{\mu ,\lambda }^{\prime }\left(w_{\lambda }\right)\left(w_{\lambda }\right)=0$$

This leads to the conclusion that

$$\mathcal{A}\left(w_{\lambda }\right)-\lambda {\int }_{\Omega }f(x,w_{\lambda }\left(x\right))w_{\lambda }\left(x\right)\mathrm{d}x=0,\mathrm{for}\mathrm{any}\lambda \in ]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[.$$

(9)

As a result, using (8) and (9), we deduce that

$$\underset{\lambda \to 0^{+}}{lim}\mathcal{A}\left(w_{\lambda }\right)=0.$$

Moreover, by Remark 2, we have

$$[\parallel w_{\lambda }{\parallel ]}_p\le \tilde{M}\mathcal{A}\left(w_{\lambda }\right)\mathrm{for}\mathrm{all}\lambda \in ]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[.$$

Thus, we can conclude that

$$\underset{\lambda \to 0^{+}}{lim}\parallel w_{\lambda }\parallel =0.$$

Furthermore, since the functional $I_{\mu ,\lambda }$ attains a global minimum over ${\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)$, which is also a local minimum of $I_{\mu ,\lambda }$ that belongs in X, it follows that the function

$$h\left(\lambda \right):=I_{\mu ,\lambda }\left(w_{\lambda }\right)$$

remains negative for $\lambda \in ]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[$, given that $w_{\lambda }\ne 0$ and $I_{\mu ,\lambda }\left(0\right)=0$.

In the sequel, let $w_{{\lambda }_1}$ and $w_{{\lambda }_2}$ be critical points of $I_{\mu ,\lambda }$ (local minimizers) corresponding to parameters ${\lambda }_1$ and ${\lambda }_2$ in $]0,{\tilde{\lambda }}_{\mu }\left(\gamma \right)[$, with ${\lambda }_1<{\lambda }_2$. Define

$$S_i:=\underset{w\in {\mathcal{A}}^{-1}\left(\right]-\infty ,{\left[\gamma \right]}_p\left[\right)}{inf}\left({\displaystyle \frac{\mathcal{A}\left(w\right)}{{\lambda }_i}}-\mathcal{V}\left(w\right)\right)={\displaystyle \frac{1}{{\lambda }_i}}{I}_{\mu ,{\lambda }_i}\left(w_{{\lambda }_i}\right),i=1,2.$$

Since $S_i<0$ for $i=1,2$ and given that ${\lambda }_1<{\lambda }_2$, we deduce that $S_2\le S_1$. As a result, we obtain

$$I_{\mu ,{\lambda }_2}\left(w_{{\lambda }_2}\right)={\lambda }_2{S}_2\le {\lambda }_2{S}_1<{\lambda }_1{S}_1=I_{\mu ,{\lambda }_1}\left(w_{{\lambda }_1}\right).$$

Thus, the proof is concluded, as $\lambda \in ]0,{\tilde{\lambda }}_{\mu }[$ was chosen arbitrarily.

Now, we are concerned by the case, when $p\left(x\right)=p$ and $s\left(x\right)=s$ are constants such that $1<s<p^{*}:={\displaystyle \frac{Np}{N-2p}}$, with $N\ge 3$; in this case, problem (1) becomes

$$\left\{\begin{array}{cc}{\Delta }_p^{2}w-{\Delta }_{p}w+\eta \left(x\right){\left|w\right|}^{p\left(x\right)-2}w=\mu \theta \left(x\right){\displaystyle \frac{{\left|w\right|}^{p-2}}{\delta {\left(x\right)}^{2p}}}+\lambda f(x,w),\hfill & \mathrm{in}\mathcal{D},\hfill \\ \\ w=\Delta w=0,\hfill & \mathrm{on}\partial \mathcal{D},\hfill \end{array}\right.$$

(10)

where $\mathcal{D}$ is a bounded domain of ${\mathbb{R}}^N$, with the regular boundary $\partial \mathcal{D}$; furthermore,

$$\tilde{M}:={\displaystyle \frac{p{C}_H}{C_H-\mu {\left|\theta \right|}_{\infty }}},K:={\displaystyle \frac{C_H}{{\left|\theta \right|}_{\infty }}},C_H={\left(N(p-1)(N-2p)/p^2\right)}^p,$$

and

$${\tilde{\lambda }}_{\mu }:=s\underset{\gamma >0}{sup}{\displaystyle \frac{{\gamma }^{p-1}}{s{c}_s{\left|M_1\left(x\right)\right|}_{\frac{s}{s-1}}{\tilde{M}}^{\frac{1}{p}}+c_s^s{M}_2{\tilde{M}}^{\frac{s}{p}}{\gamma }^{s-1}}}.$$

In what follows, denoted by $A_1=s{c}_s{\left|M_1\left(x\right)\right|}_{{\displaystyle \frac{s}{s-1}}}{\tilde{M}}^{{\displaystyle \frac{1}{p}}}$ and by $A_2=c_s^s{M}_2{\tilde{M}}^{{\displaystyle \frac{s}{p}}}$, direct computations give

$${\tilde{\lambda }}_{\mu }=\left\{\begin{array}{cc}+\infty \hfill & \mathrm{for}1<s<p,\hfill \\ {\displaystyle \frac{C_H-\mu {\left|\theta \right|}_{\infty }}{c_p^p{M}_2{C}_H}}\hfill & \mathrm{for}s=p,\hfill \\ {\displaystyle \frac{s{\tilde{\gamma }}_{max}^{p-1}}{s{c}_s{\left|M_1\left(x\right)\right|}_{\frac{s}{s-1}}{\tilde{M}}^{\frac{1}{p}}+c_s^s{M}_2{\tilde{M}}^{\frac{s}{p}}{\tilde{\gamma }}_{max}^{s-1}}}\hfill & \mathrm{for}s\in ]p,p^{*}[,\hfill \end{array}\right.$$

where,

$${\tilde{\gamma }}_{max}:={\left({\displaystyle \frac{A_1(p-1)}{A_2(s-p)}}\right)}^{{\displaystyle \frac{1}{s-1}}}.$$

Now, by using Remark 3.4 in [2], we have an analog result as Theorem 3.5 in [2] and Theorem 3.3 in [3]. In fact, we have the following result.

Corollary 1. 

Let $f:\mathcal{D}\times \mathbb{R}\to \mathbb{R}$ be a function satisfying the Carathéodory conditions and the property $f(x,0)=0$. Assume further that f is bounded by

$$\left|f(x,t)\right|\le M_1\left(x\right)+M_2{\left|t\right|}^{s-1},\mathit{for}\mathit{almost}\mathit{every}(x,t)\in \mathcal{D}\times \mathbb{R},$$

(11)

where $s\in ]p,p^{*}[$, $M_1\left(x\right)\in L^{{\displaystyle \frac{s}{s-1}}}\left(\mathcal{D}\right)$, and $M_2$ is a positive constant.

Additionally, suppose the existence of a non-empty open subset $\Omega \subseteq \mathcal{D}$ and a measurable one ${\Omega }_1\subseteq \Omega$ such that $\mathit{meas}\left({\Omega }_1\right)>0$ and

$$\underset{t\to 0^{+}}{lim\; sup}{\displaystyle \frac{{inf}_{x\in {\Omega }_1}F(x,t)}{t^p}}=+\infty ,\mathit{and}\underset{t\to 0^{+}}{lim\; inf}{\displaystyle \frac{{inf}_{x\in \Omega }F(x,t)}{t^p}}>-\infty .$$

Under these assumptions, for any fixed $\mu \in [0,K)$ where $K:={\displaystyle \frac{C_H}{{\left|\theta \right|}_{\infty }}}$, there is a constant ${\tilde{\lambda }}_{\mu }>0$ given by

$${\tilde{\lambda }}_{\mu }:={\displaystyle \frac{s{\tilde{\gamma }}_{max}^{p-1}}{s{c}_s{\left|M_1\left(x\right)\right|}_{\frac{s}{s-1}}{\tilde{M}}^{\frac{1}{p}}+c_s^s{M}_2{\tilde{M}}^{\frac{s}{p}}{\tilde{\gamma }}_{max}^{s-1}}},$$

where

$${\tilde{\gamma }}_{max}:={\left({\displaystyle \frac{A_1(p-1)}{A_2(s-p)}}\right)}^{{\displaystyle \frac{1}{s-1}}},\mathit{and}\tilde{M}:={\displaystyle \frac{p{C}_H}{C_H-\mu {\left|\theta \right|}_{\infty }}},$$

such that for any $\lambda \in ]0,{\tilde{\lambda }}_{\mu }[$, problem (10) admits at least one nontrivial weak solution $w_{\lambda }\in X$, where $X=W_0^{1,p}(\Omega )\cap W^{2,p}(\Omega )$. Moreover,

$$\underset{\lambda \to 0^{+}}{lim}∥w_{\lambda }∥=0,$$

Additionally, the function $\lambda \mapsto I_{\lambda ,\mu }\left(w_{\lambda }\right)$ remains negative and strictly decreases over the interval $]0,{\tilde{\lambda }}_{\mu }[$.

Remark 3. 

Since, in the constant case, the operator involved in our work combines both the p-Laplacian and the p-biharmonic operators, and taking into account that $M_1\left(x\right)\in L^{{\displaystyle \frac{s}{s-1}}}\left(\mathcal{D}\right)$, the previous corollary is a natural generalization of Theorem 3.5 in [2] and Theorem 3.3 in [3].

In what follows, and in order to illustrate an example for Corollary 1, we assume that $L^{{\displaystyle \frac{s}{s-1}}}\left(\mathcal{D}\right)\ni M_1\left(x\right)=M_1$ is a positive constant; then, $A_1$ becomes ${\tilde{A}}_1=c_s{\left|\mathcal{D}\right|}^{{\displaystyle \frac{s-1}{s}}}{\tilde{M}}^{{\displaystyle \frac{1}{p}}}$ and

$${\tilde{\lambda }}_{\mu }={\displaystyle \frac{s{\tilde{\gamma }}_{max}^{p-1}}{s{c}_s{M}_1{\left|\mathcal{D}\right|}^{\frac{s-1}{s}}{\tilde{M}}^{\frac{1}{p}}+c_s^s{M}_2{\tilde{M}}^{\frac{s}{p}}{\tilde{\gamma }}_{max}^{s-1}}},fors\in ]p,p^{*}[,$$

where,

$${\tilde{\gamma }}_{max}:={\left({\displaystyle \frac{{\tilde{A}}_1(p-1)}{A_2(s-p)}}\right)}^{{\displaystyle \frac{1}{s-1}}}.$$

The following example is inspired from [2].

4. Example

Assume that $f(x,t):={\xi \left(x\right)\left|t\right|}^{r-2}u+\eta \left(x\right){\left|t\right|}^{s-2}u$ and $\xi ,\eta :\mathcal{D}\to \mathbb{R}$ are two continuous positive and bounded functions, where $1<r<p$ and $p<s<p^{*}$. It is obvious that the nonlinearity f vanishes at 0; moreover,

$$\left|f(x,t)\right|⩽2max\left\{{\parallel \xi \parallel }_{\infty },{\parallel \eta \parallel }_{\infty }\right\}\left(1+{\left|t\right|}^{s-1}\right),\mathrm{for} \mathrm{a}.\mathrm{e}(x,t)\in \mathcal{D}\times \mathbb{R},$$

then condition (11) is achieved for $M_1\left(x\right)=M_1=M_2=2max\left\{{\parallel \xi \parallel }_{\infty },{\parallel \eta \parallel }_{\infty }\right\}$.

Furthermore, one has

$$\underset{\tau \to 0^{+}}{lim}{\displaystyle \frac{{\int }_{\mathcal{D}}F(x,\tau )dx}{{\tau }^p}}=+\infty ,$$

consequently, for any $\mu \in [0,{\displaystyle \frac{C_H}{{\left|\theta \right|}_{\infty }}}[$, and

$$\lambda \in \left]0,{\displaystyle \frac{s{\tilde{\gamma }}_{max}^{p-1}}{2M_1\left(s{c}_s{\left|\mathcal{D}\right|}^{\frac{s-1}{s}}{\tilde{M}}^{\frac{1}{p}}+c_s^s{\tilde{M}}^{\frac{s}{p}}{\tilde{\gamma }}_{max}^{s-1}\right)}}\right[,$$

where ${\tilde{\gamma }}_{max}:={\left({\displaystyle \frac{{\tilde{A}}_1(p-1)}{A_2(s-p)}}\right)}^{{\displaystyle \frac{1}{s-1}}}$, problem (10) possesses one nontrivial weak solution ${\tilde{w}}_{\lambda }\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$; furthermore,

$$\underset{\lambda \to 0^{+}}{lim}∥{\tilde{w}}_{\lambda }∥=0.$$