Triple solutions for a Leray-Lions $p(x)$-biharmonic operator involving Hardy potential and indefinite weight

1.   Introduction

Honoring Jean Leray and Jacques-Louis Lions [1], the Leray-Lions operator is an essential component of nonlinear partial differential equations (PDEs) that emerge in a variety of scientific and engineering fields such as electro-rheological fluids (Ružička [2]), elastic mechanics (Zhikov [3]), stationary thermo-rheological viscous flows of non-Newtonian fluids (Rajagopal-Ružička [4]), image processing (Chen-Levine-Rao [5]), and so on. For the purpose of examining the existence, uniqueness, regularity, and multiplicity of solutions to certain equations, this operator offers a strict framework.

The Leray-Lions operator gains more intricacy and richness when a Hardy potential is incorporated. The Hardy potential introduces singularities around the origin, which makes the operator's characteristics and the behavior of the solutions more delicate.

In this work, we will establish the existence of a triple weak solution to the following intriguing problem.

where $g(x,u)=f(x)|u|^{r(x)-2}u$, $\Omega$ is a bounded domain in ${{\mathbb R}}^{N}$ ($N\geq 3$), with boundary $\partial\Omega$ of class $C^1$, $s$ is a constant, $\lambda>0$ is a parameter, $f$ is a function in a Lebesgue generalized space $L^{\gamma(x)}(\Omega)$, and functions $r,p$, and $\gamma \in C(\overline\Omega)$ verify the following inequalities:

for all $x\in\Omega,$ where $\Delta\left(a(x, \Delta u)\right)$ is the fourth-order Leray-Lions operator, and $a$ is a Carathéodory function that satisfies an appropriate additional requirements.

The study of higher-order, nonlinear PDEs with variable exponents is crucial as it can provide insights into the behavior of complex physical systems that exhibit nonlinear and anisotropic properties. Moreover, the inclusion of the fourth-order Leray-Lions operator $\Delta(a(x, \Delta u))$ in the problem adds an extra layer of complexity, making it an interesting and challenging problem. While the theory of variable exponent Lebesgue and Sobolev spaces has been extensively developed in the literature, the establishment of the existence of a triple weak solution to the given problem remains an open and non-trivial task. Previous studies have tackled similar problems, but the combination of the higher-order operator, the nonlinear terms, and the variable exponents introduces new challenges that require the development of specialized techniques and tools. This study aims to fill the existing research gap by establishing the existence of a triple weak solution to the Problem (1.1). To achieve this, the authors will employ advanced techniques from nonlinear analysis, such as variational methods, critical point theory, and fixed point theorems, along with a deep understanding of variable exponent Lebesgue and Sobolev spaces. The proposed approach will involve a careful analysis of the equation, the boundary conditions, and the parameter dependence, leading to the development of a comprehensive framework for the existence of a triple weak solution. The novelty and originality of this work lie in the fact that it combines the challenging aspects of higher-order PDEs, nonlinear terms, and variable exponents, which have not been fully addressed in the existing literature. The successful resolution of this problem will contribute to the advancement of the theory of nonlinear PDEs with variable exponents and may have further applications in various scientific and engineering disciplines. The main difficulties in establishing the existence of a triple weak solution to the given fourth-order, nonlinear problem with variable exponents lie in the inherent complexity of the equation. The presence of the higher-order Leray-Lions operator, the nonlinear terms with variable exponents, and the homogeneous Dirichlet boundary conditions on both the function and its Laplacian pose significant challenges. Addressing these challenges requires the development and application of advanced techniques from nonlinear analysis, such as variational methods, critical point theory, and fixed point theorems, coupled with a thorough understanding of variable exponent Lebesgue and Sobolev spaces. Furthermore, the parameter dependence of the problem adds an additional layer of complexity that must be carefully handled in the analysis. Recently, (Liu-Zhao [6]) established an existence and multiplicity result for Problem (1.1). Under certain appropriate conditions on nonlinearity $g$, we mention that our paper is an extension of that of Liu-Zhao; in fact, the nonlinearity $g$ may change sign on $\Omega$. Moreover the condition of type Ambrosetti-Rabinowitz is not needed. This paper is structured as follows: We give some background and preliminaries on the Sobolev spaces with variable exponents in Section 2, and Sections 3 and 4 include the proof of our results.

2.   Background and preliminaries

Throughout this paper, let

Put

Moreover, for $\beta\in C_{+}(\overline{\Omega})$, put

which yields to $[\kappa]^{\frac{1}{\beta}}:=\max\{\kappa^{\frac{1}{\beta^+}}, \kappa^{\frac{1}{\beta^-}}\}$ and $[\kappa]_{\frac{1}{\beta}}:=\min\{\kappa^{\frac{1}{\beta^+}}, \kappa^{\frac{1}{\beta^-}}\}$.

In this study, we assume that $\Omega \subset \mathbb{R}^{N}(N>2)$ is a bounded regular domain with $C^{1}$ boundary, $p, r, \gamma\in C_{+}(\overline{\Omega})$, which holds the following:

We denote the variable exponent Lebesgue space [7] by

We define its Luxemburg norm as

For every $u \in L^{p(x)}(\Omega)$ and $v \in L^{p^{\prime}(x)}(\Omega)$, where $L^{p^{\prime}(x)}(\Omega)$ is the conjugate space of $L^{p(x)}(\Omega)$, one has the following Hölder type inequality

Let's now recall the following proposition.

Proposition 2.1 ([8]). For every $u \in L^{p(x)}(\Omega)$, one has

Proposition 2.2 ([9]). Let $p$ and $q$ be two functions in $C_{+}(\overline{\Omega})$ satisfying $q(x) \leq p(x)$, a.e. on $\Omega$, then $L^{p(x)}(\Omega) \hookrightarrow L^{q(x)}(\Omega)$, and

where $c_q$ is a positive constant.

For for $k\in\{1,2\}$, the Sobolev space with variable exponent $W^{k, p(x)}(\Omega)$ is defined as

where $\alpha=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{N}\right)$ is a multi-index such that $|\alpha|= \Sigma_{i=1}^{N} \alpha_{i}$, $D^{\alpha} u=\frac{\partial^{|\alpha|}u}{\partial^{\alpha_1} x_{1} \ldots \partial^{\alpha_N} x_{N}}$. The above space endowed with the norm

is a reflexive separable Banach space. Let $W_{0}^{1, p(x)}(\Omega)$ be the closure of $C_{0}^{\infty}(\Omega)$ in $W^{1, p(x)}(\Omega)$, which has the norm $\|u\|_{1, p(x)}=|D u|_{p(x)}$. In the following, let

endowed with the norm

The modular on $X$ is the mapping $\rho_{p(x)} : X \rightarrow \mathbb{R}$ defined by $\rho_{p(x)}(u)=\int_{\Omega}|\Delta u|^{p(x)}dx$. This mapping meets the same characteristics as Proposition 2.3. To be more specific, we have the following.

Proposition 2.3. For every $u\in L^{p(x)}(\Omega)$, one has

(1) $\|u\|<1 \ (resp, =1, >1) \Leftrightarrow \rho_{p(x)}(u)<1\ (resp, =1, >1),$

(2) $[\|u\|]_p\leq\rho_{p(x)}(u)\leq [\|u\|]^p.$

Proposition 2.4 ([10]). Let $p$ and $q$ be measurable functions such that $p\in L^{\infty}(\Omega)$, and $1\leq p(x)q(x)\leq\infty$, for a.e. $x\in\Omega$. Let $w\in L^{q(x)}(\Omega)$, $w\neq0$. Then

Remember that the critical Sobolev exponent is defined as:

As a result of Proposition 2.2, if $q(x) \leq p(x)$ a.e on $\Omega$, we have

In particular, one has

The following definitions and assertions will be required in Section 3.

Definition 2.1. Let $\Phi$ and $\Psi$ be two continuously Gâteaux differentiable functionals on a real Banach space $X$ and let $d \in \mathbb{R}$. The functional $I:=\Phi-\Psi$ verifies the Palais-Smale condition cut of upper at $d$ ($(P S)^{[d]}$ ) if any sequence $\left\{u_{n}\right\}_{n \in \mathbb{N}} \in X$, which verifies

● $I\left(u_{n}\right)$ is bounded,

● $\lim _{n \rightarrow +\infty}\left\|I^{\prime}\left(u_{n}\right)\right\|_{X^{*}}=0$,

● $\Phi\left(u_{n}\right)<d$ for each $n \in \mathbb{N}$,

has a convergent subsequence.

If $d=\infty$, the functional $I:=\Phi-\Psi$ fulfills the Palais-Smale condition.

Our main existence result is due to the following theorem.

Theorem 2.1. [11], Theorem 3.1] Let $X$ be a real Banach space, and let $\Phi, \Psi: X \rightarrow \mathbb{R}$ be two continuously Gâteaux differentiable functionals such that

Suppose that there is a positive constant $d \in \mathbb{R}$ and $\overline{x} \in X$ with $0<\Phi(\overline{x})<d$ such that

and for any

$I_{\lambda}=\Phi-\lambda \Psi$ fulfills the $(P S)^{[d]}$-condition, so for every $\lambda \in \Lambda$, there is $x_{\lambda} \in$ $\Phi^{-1}(] 0, d])$ such that $I_{\lambda}\left(x_{\lambda}\right) \leq I_{\lambda}(x)$ for all $x \in \Phi^{-1}(] 0, d])$ and $I_{\lambda}^{\prime}\left(u_{\lambda}\right)=0$.

The multiplicity result is due to the following theorem.

Theorem 2.2 ([12]). Let $X$ be a reflexive real Banach space, $\Phi: X \rightarrow \mathbb{R}$ be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on $X^{*}$ and $\Psi: X \rightarrow \mathbb{R}$ be a continuously Gâteaux differentiable whose Gâteaux derivative is compact such that

Suppose that there exist $d>0$ and $\overline{x} \in X$, with $d<\Phi(\overline{x})$, such that

$(a_2)$ for each $\left.\lambda \in \Lambda_{d}:=\right] \frac{\Phi(\overline{x})}{\Psi(\overline{x})}, \frac{d}{\sup _{\Phi(x)\leq d} \Psi(x)}[,$ $I_{\lambda}:=\Phi-\lambda \Psi$ is coercive.

Then, for any $\lambda \in \Lambda_{d}$, $\Phi-\lambda \Psi$ has at least three distinct critical points in $X$.

Let $\delta(x):=\sup\{\delta>0 \mid B(x,\delta)\subseteq\Omega\}$, for all $x\in\Omega$ where $B$ is the ball centered at $x$ and of radius $\delta$. We can see easily that there exists $x^0\in\Omega$ such that $B(x^0,R)\subseteq\Omega$, where $R= \sup\limits_{x\in\Omega}\delta(x).$ This paper will require the following hypotheses:

$A: \overline{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function with continuous derivative $a(x, \xi)=\partial_{\xi} A(x, \xi)$, satisfying $a(x, u+v) \leq$ $c(a(x, u)+a(x, v)), \forall u, v \in W_{0}^{1, p(x)}(\Omega)\cap W^{2,p(x)}(\Omega)$, for some positive constant $c$, and $A$ satisfies the following assumptions:

$(A1)$ $A(x, 0)=0, A(x, \xi)=A(x,-\xi)$ for all $x \in \Omega, \xi \in \mathbb{R}$;

$(A2)$ $|a(x, \xi)| \leqslant c_1\left(\alpha(x)+|\xi|^{p(x)-1}\right)$ a.e. $(x, \xi) \in \Omega \times \mathbb{R}$, where $c_1 > 0, \alpha(x) \in L^{\frac{p(x)}{p(x)-1}}(\Omega), 1<p(x) \in C(\overline{\Omega})$;

$(A3)$ $|\xi|^{p(x)} \leq a(x, \xi) \cdot \xi \leq p(x) A(x, \xi)$ for a.e. $x \in \Omega$ and all $\xi \in \mathbb{R}$;

$(A4)$ Assume that $f\in L^{\gamma(x)}(\Omega)$ satisfies the following:

where $B(x^0,R)$ is the ball of radius $R$ centered at $x^0$ and $f_0$ is a positive constant.

Remark 2.1. By $(A2)$, one has

for some constant $C>0$.

Throughout this work, we will denote by $m:= \frac{\pi^{\frac{N}{2}}}{\frac{N}{2}\Gamma(\frac{N}{2})},$ where $\Gamma$ is the Gamma function.

3.   Existence

This section is devoted to show some reacquired results needed to establish the existence and multiplicity result. We start by recalling the following remark.

Remark 3.1 (Kefi [13]). The conjugate exponent of the function $\gamma(x)$ will be denoted by $\gamma'(x)$ and $\beta(x):= \frac{\gamma(x)r(x)}{\gamma(x)-r(x)}.$ There are compact and continuous embeddings $X \hookrightarrow L^{\gamma'(x)r(x)}(\Omega)$ and $X \hookrightarrow L^{\beta(x)}(\Omega)$, as well as the best constant $k>0$ such that

In what follows, we recall the Hardy-Rellich inequality [14].

Lemma 3.1. For $1<s<N / 2$ and $u \in W_{0}^{1, s}(\Omega) \cap W^{2, s}(\Omega)$, we have

where

Let's review what a weak solution to Problem $(1.1)$ is

Definition 3.1. $u\in X\backslash\{0\}$ is a weak solution of Problem $(1.1)$ if $\Delta u=0$ on $\partial{\Omega}$ and

Let us denote by

The Euler-Lagrange functional for Problem $(1.1)$ is thus defined as $I_{\lambda}: X \rightarrow \mathbb{R},$

where

It is obvious that assertion $(a_0)$ of Theorem 2.2 holds, moreover Remark 2.1 assures that $\Phi$ is well-defined and by virtue of Proposition 2.4 and Remark 3.1, $\Psi$ it the same as well, since we have for all $u\in X,$

Furthermore, by inequality (3.1) in Remark 3.1, one has

As a result, $\Psi$ is a well-defined and

for all $u, v\in X$ is compact (see [15]). Moreover, by using Proposition 2.3 and hypothesis $(A3)$ for $u\in X$ with $\|u\| >1,$ one has

and consequently, $\Phi$ is coercive. On the other hand, $\Phi$ is sequentially weakly lower semi-continuous, and of class $C^1$ on $X$ (Liu-Zhao [6]), with

Moreover, we have the following.

Proposition 3.1. $\Phi': X \rightarrow X^{*}$ is coercive and uniformly monotone and admits a continuous inverse in $X^*$.

Proof. For the coercivity and due to assertion $(A3)$ and Proposition 2.3, one has for any $u\in X$ with $\|u\|\geq 1$,

so $\Phi'$ is coercive.

Using the assumption on $\theta$, one has

Now, let $U_{\beta}=\{x\in\Omega:\beta(x)\geq 2\}$ and $V_{\beta}=\{x\in\Omega:1<\beta(x)<2\}$. By using the elementary inequality [16], for $\beta>1$ there exists a positive constant $C_{\beta},$ such that if $\beta \geq 2$, then

and if $1<\beta<2$, then

where $\langle\cdot, \cdot\rangle$ denotes the usual inner product in $\mathbb{R}^{N}$. Due to the fact that for any $u, v \in X$,

for some $c>0$, and by assumptions (A1) and (A3), we have

By Proposition 2.3 and taking into account inequalities (3.3)–(3.5), one has, for any $u, v \in X$,

which assures that $\Phi'$ is uniformly monotone. Theorem 26.(A)d of [17] ends the proof. □

Proposition 3.2. $\Phi'$ satisfies the condition $(S)_+$, which means: If $u_n\rightharpoonup u$ and

then $u_n\rightarrow u$ (strongly).

Proof. Since $u_n\rightharpoonup u$ and $\overline{lim}_{n\rightarrow +\infty}\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle\leq 0$, then due to $(3.6)$, one has

which ends the proof. □

Remark 3.2. Under assumptions $(A2)$ and $(A3)$, one has

where

Proof. By using assumptions $(A2)$ and $(A3)$ and Proposition 2.3, we have

By Hardy's inequality, we deduce

where

and this ends the proof. □

Remark 3.3. If $I_{\lambda}^{\prime}(u)=0$, we have

for any $u, v \in X$, which assures that the critical points of $I_{\lambda}$ are exactly weak solutions of Problem $(1.1)$.

Lemma 3.2. $I_{\lambda}$ fulfills the Palais-Smale condition for any $\lambda>0$.

Proof. Let $\left\{u_{n}\right\} \subseteq X$ be a Palais-Smale sequence, so, one has

Let us show that $\left\{u_{n}\right\} \subseteq X$ contains a convergent subsequence. By the Hölder inequality, Proposition 2.4 and Remark 3.1, we have

So, for $n$ large enough, by assumption $(A3)$ and Proposition 2.3, one has

Moreover, using (3.7), we have

since $r^+<p^-$, then $\left\{u_{n}\right\}$ is bounded and passing to a subsequence if necessary, we can assume that $u_{n} \rightharpoonup u$. By Proposition 3.2, $u_{n}\rightarrow u$ (strongly) in $X$, and so $I_{\lambda}$ fulfills the Palais-Smale condition. □

Our existence result is the following:

Theorem 3.1. Suppose that there exist $d,\delta>0$ such that

where $m:=\frac{\pi^{N / 2}}{N / 2 \Gamma(N / 2)}$ is the measure of the unit ball of $\mathbb{R}^{N}$, and $\Gamma$ is the Gamma function. So, for any $\lambda \in] A_{\delta}, B_{d}[$, with

and

and Problem (1.1) has at least one nontrivial weak solution.

Proof. We try to prove our existence result using Theorem 2.1. For this purpose, we have to show that all conditions of Theorem 2.1 are met.

To begin, for a given $\lambda>0$, we mention that from Lemma 3.2, the functional $I_{\lambda}$ fulfills the $(P S)^{[d]}$ condition. Let $d>0, \delta>0$ be as in $(3.8)$ and let $w \in X$ be defined by

where $x=\left(x_{1}, \ldots, x_{N}\right) \in \Omega$. Then,

So, by applying Remark 3.2, one has

so, $\Phi(w)<d$. On the other hand, one has

then, we deduce that

Using Remark 2.3, for any $u \in \Phi^{-1}((-\infty, d])$, we have

Hence, from Proposition 2.4 and Remark 3.2, we deduce

So

As a result, the criteria of Theorem 2.1 are confirmed. So, for any

$I_{\lambda}$ admits at least one nonzero critical point, which is the problem's weak solution. □

4.   Triple solutions

Theorem 4.1. For any $\lambda \in] A_{ \delta}, B_{d}\left[\right.$, $A_{\delta}$ and $B_{d}$ are those of Theorem 3.1,

and

and Problem (1.1) admits at least three weak solutions.

Proof. Note that $\Phi$ and $\Psi$ fulfill the regularity assumptions of Theorem 2.2. Let us verify conditions $(i)$ and $(ii)$ of this theorem. For this purpose, let

and let $w \in X$ be as mentioned above, that is,

So, by applying assumption $(A3)$ and Remark 3.2, one has

Therefore, the assumption $(i)$ of Theorem 2.2 holds. Let us show that $I_{\lambda}$ is coercive for any $\lambda>0$. By using (3.10), one has

additionally, from Remark 3.2,

So,

and by using $p^{-}>r^{+}>1$, we deduce that $I_{\lambda}$ is coercive and, consequently, condition (ⅱ) is fulfilled, which assures that all assumptions of Theorem 4.1 are satisfied. So, for any $\lambda \in] A_{ \delta}, B_{d}\left[\right.$, $I_{\lambda}$ has at least three distinct critical points, which represents the weak solutions of Problem (1.1). □

Example 4.1. Let $\Omega = B(0,1) \subset \mathbb{R}^3$ be the unit ball and put $s$ such that $1<s<\frac{11}{10}$ and $a(x,t)=\vartheta(x)|t|^{p(x)-2}t$, $\vartheta\equiv 1$; moreover, let $\gamma(x)=\gamma$ be a constant such that $\gamma>\frac{3}{2}$ and define the variable exponents and weight function as,

It's obvious that

Problem (1.1) can be written as

and admits three distinct weak solutions, one of which may be the trivial one.

5.   Conclusions

The paper focuses on the existence and multiplicity of three weak solutions for a Leray-Lions $p(x)$-biharmonic problem involving constant Hardy potential. We may extend the obtained results to the existence of generalized solutions to Kirchhoff type elliptic equations involving variable Hardy singular coefficients such as the following sixth-order variable exponent Kirchhoff type $p(x)$-triharmonic equations involving $(h,r(x))$-Hardy singular coefficients:

Furthermore, we can reduce the regularity of corresponding weighted functions $a(x),b(x),c(x).$

Author contributions

Conceptualization: Kefi formulated the initial research problem and developed the overarching mathematical framework. Validation: Liu independently verified the correctness of the mathematical results. Both authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FPEJ-2024-1706-02. This work is supported by Natural Science Foundation of Shandong Province, China (ZR2021MA070).

Conflict of interest

The authors declare no conflicts of interest.