Weak Solutions to Leray–Lions-Type Degenerate Quasilinear Elliptic Equations with Nonlocal Effects, Double Hardy Terms, and Variable Exponents

Abstract

This study investigates the existence and multiplicity of weak solutions for a class of degenerate weighted quasilinear elliptic equations that incorporate nonlocal nonlinearities, a double Hardy term, and variable exponents. The problem encompasses a degenerate nonlinear operator characterized by variable exponent growth, along with a nonlocal interaction term and specific constraints on the nonlinearity. By employing critical point theory, we establish the existence of at least three weak solutions under sufficiently general assumptions.

1. Introduction

Elliptic equations incorporating Leray–Lions-type degenerate operators, Hardy-type singularities, and nonlocal nonlinearities play a crucial role in modeling diverse phenomena across physics, biology, and engineering. The degeneracy in the operator extends classical elliptic models, enabling the study of heterogeneous media, while Hardy-type potentials introduce singular terms relevant in quantum mechanics and fluid dynamics. Additionally, nonlocal nonlinearities capture long-range interactions in applications such as population dynamics, image processing, and nonlinear elasticity, further complicating the mathematical analysis. The interplay of these factors presents significant challenges in proving the existence, uniqueness, and multiplicity of solutions, necessitating sophisticated variational techniques. Given their relevance in porous media flow, plasma physics, and financial modeling, investigating such equations is essential both from a theoretical and applied perspective.

The analysis of elliptic equations becomes significantly more intricate due to the presence of singularities and degeneracies, which profoundly influence the behavior of solutions. Singularities, particularly those located at the origin or on the boundary, can substantially modify the properties of the differential operator, making the solutions highly sensitive to variations in the domain. For instance, when $1<p<N$, it is established that if $u\in W^{1,p}\left({\mathbb{R}}^N\right)$, then $u/\left|x\right|\in L^p\left({\mathbb{R}}^N\right)$, and similarly, if $u\in W^{1,p}\left(D\right)$, then $u/\left|x\right|\in L^p\left(D\right)$, where D is a bounded domain (see Lemma 2.1 in [1]). This framework naturally gives rise to Hardy-type inequalities, which play a crucial role in regulating the singular behavior of solutions near critical points, especially when the equation incorporates singular potential terms (see, e.g., [1,2,3,4,5,6,7,8,9,10,11]).

In addition to the challenges posed by singularities, the presence of nonlocal terms further increases the complexity of the problem. These terms, often originating from integral formulations or global coupling effects, introduce dependencies that impact the solution both locally and throughout the entire domain. Due to this nonlocal influence, standard analytical methods may not suffice, requiring more advanced techniques to establish the existence and multiplicity of solutions.

Moreover, the degeneracy of the operator presents significant challenges, particularly when it involves a weighted function $\kappa \left(x\right)$ in the context of the p-Laplacian or $p\left(x\right)$-Laplacian. The nature of $\kappa \left(x\right)$, whether singular or merely bounded, plays a crucial role in determining the appropriate functional framework. In such cases, the classical Sobolev spaces $W^{1,p}\left(D\right)$ and $W^{1,p\left(x\right)}\left(D\right)$ may no longer be suitable for capturing the problem’s structural complexities. To effectively address the impact of degeneracies and singularities, it becomes necessary to adopt weighted Sobolev spaces such as $W^{1,p}(\kappa ,D)$, which provide a more refined analytical setting (see [12] for further details). In the case of parabolic equations with a degenerate operator, one can refer to the recent work [13] for further insights.

This study investigates a class of weighted quasilinear elliptic equations characterized by nonlocal nonlinearities, a double Hardy term, and variable exponents. More precisely, we investigate the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-divb(x,\nabla u)+{\displaystyle \frac{{\eta \left(x\right)\left|u\right|}^{q-2}u}{{\left|x\right|}^q}}+{\displaystyle \frac{{c\left(x\right)\left|u\right|}^{g\left(x\right)-2}u}{{\left|x\right|}^{\gamma \left(x\right)}}}=\lambda f(x,u){\left({\int }_{D}F(x,u)dx\right)}^r\hfill & \mathrm{in}D,\hfill \\ u=0\hfill & \mathrm{on}\partial D,\hfill \end{array}\right.\end{array}$$

(1)

here, D denotes a bounded domain in ${\mathbb{R}}^N$ with a Lipschitz boundary $\partial D$. The function $b:D\times {\mathbb{R}}^N\to {\mathbb{R}}^N$ represents a nonlinear operator, while $\eta \left(x\right)$, $c\left(x\right)$ are non-negative functions belonging to $L^{\infty }\left(D\right)$. The function $f:D\times \mathbb{R}\to \mathbb{R}$ is assumed to be a Carathéodory condition. Additionally, $\gamma \left(x\right)$ is a function that belongs to $C\left(\overline{D}\right)$ and takes values in the range $[0,N)$, r is a positive constant, and $\lambda$ is a positive parameter. The operator $divb(x,\nabla u)$ extends the degenerate $p\left(x\right)$-Laplacian

$$div\left({\kappa \left(x\right)|\nabla u|}^{p\left(x\right)-2}\nabla u\right),$$

where $p\left(x\right)\in C(\overline{D},(1,\infty )):=C_{+}\left(\overline{D}\right)$ and $\kappa \left(x\right)\in S_{+}\left(D\right)$, representing the class of all measurable functions on D that remain positive almost everywhere in D. For any function $h\left(x\right)\in C_{+}\left(\overline{D}\right)$, we define its minimum and maximum values as

$$h^{-}:=\underset{x\in \overline{D}}{min}h\left(x\right),h^{+}:=\underset{x\in \overline{D}}{max}h\left(x\right).$$

Additionally, we denote by $h^{\prime }$ the conjugate function of h, which satisfies the identity

$${\displaystyle \frac{1}{h\left(x\right)}}+{\displaystyle \frac{1}{h^{\prime }\left(x\right)}}=1,\forall x\in \overline{D}.$$

Under these assumptions, we establish the following conditions:

(${\mathsf{B}}_0$)

$b(x,-\xi )=-b(x,\xi )$ for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$.

(B₁)

There is a Carathéodory function $\mathcal{B}:D\times {\mathbb{R}}^N\to \mathbb{R}$, which is continuously differentiable with respect to its second argument, such that:

–

$\mathcal{B}(x,0)=0$ for almost every $x\in D$.

–

$b(x,\xi )={\nabla }_{\xi }\mathcal{B}(x,\xi )$ for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$.

(B₂)

There is a constant ${\mathcal{C}}_1>0$ such that:

$$\left|b(x,\xi )\right|\le {\mathcal{C}}_1\kappa \left(x\right)\left[\nu \left(x\right)+{\left|\xi \right|}^{p\left(x\right)-1}\right]$$

for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$, where $\nu \in S_{+}\left(D\right)$ and $\kappa \nu \in L^{p^{\prime }\left(x\right)}\left(D\right)$. The notation $|·|$ refers to the Euclidean norm.

(B₃)

The function $b(x,\xi )$ satisfies the strict monotonicity condition:

$$0<\left[b\right(x,\xi )-b(x,\zeta \left)\right]·(\xi -\zeta )$$

for almost every $x\in D$ and for all $\xi ,\zeta \in {\mathbb{R}}^N$ with $\xi \ne \zeta$.

(B₄)

There is a constant ${\mathcal{C}}_2>0$ such that:

$${\mathcal{C}}_2\kappa \left(x\right){\left|\xi \right|}^{p\left(x\right)}\le b(x,\xi )·\xi$$

for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$.

(B₅)

The function $b(x,\xi )$ satisfies the following upper bound:

$$b(x,\xi )·\xi \le p^{+}\mathcal{B}(x,\xi )$$

for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$.

(w)

The weight function $\kappa$ satisfies $\kappa \in L_{\mathrm{loc}}^1\left(D\right)$ and ${\kappa }^{-h}\in L^1\left(D\right)$, where $h\in C\left(\overline{D}\right)$ and

$$h\left(x\right)\in \left({\displaystyle \frac{N}{p\left(x\right)}},\infty \right)\cap \left[{\displaystyle \frac{1}{p\left(x\right)-1}},\infty \right)$$

for all $x\in \overline{D}$.

Assumption $\left(w\right)$ guarantees the fundamental properties of the weighted variable exponent Sobolev spaces $W^{1,p\left(x\right)}(\kappa ,D)$, which will be introduced later. This condition allows the weight function $\kappa$ to be unbounded or approach zero, leading to what is known as a degenerate problem. Moreover, it is important to emphasize that conditions $\left(B_0\right)$ through $\left(B_5\right)$ do not necessarily hold simultaneously. A typical example of operators satisfying the previous assumptions are

$$div\left[\kappa \left(x\right)\sum _{i=1}^n{|\nabla u|}^{p_i\left(x\right)-2}\nabla u\right]\mathrm{and}div\left[\kappa \left(x\right){\left(1+{|\nabla u|}^2\right)}^{\left(p\right(x)-2)/2}\nabla u\right]$$

Additionally, under assumption $\left(B1\right)$, one has for almost every $x\in D$ and for all $\xi \in {\mathbb{R}}^N$,

$$\mathcal{B}(x,\xi )={\int }_0^{\xi }b(x,t)dt.$$

(2)

From assumption $\left(B_2\right)$, it directly follows that

$$\left(\tilde{B}2\right)\mathcal{B}(x,\xi )\le {\tilde{C}}_1\left[\tilde{\nu }\left(x\right)\left|\xi \right|+{\kappa \left(x\right)\left|\xi \right|}^{p\left(x\right)}\right]\mathrm{for}\mathrm{almost}\mathrm{every}x\in D\mathrm{and}\mathrm{for}\mathrm{all}\xi \in {\mathbb{R}}^N,$$

where $\tilde{\nu }:=\kappa \nu \in L^{p^{\prime }\left(x\right)}\left(D\right)\mathrm{and}{\tilde{C}}_1\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{constant}.$

As a direct consequence of assumption $\left(B_4\right)$, we obtain

$$\left(\tilde{B}4\right)\mathcal{B}(x,\xi )\ge {\displaystyle \frac{\overline{C}\kappa \left(x\right)}{p\left(x\right)}}{\left|\xi \right|}^{p\left(x\right)}\mathrm{for}\mathrm{almost}\mathrm{every}x\in D\mathrm{and}\mathrm{for}\mathrm{all}\xi \in {\mathbb{R}}^N,$$

where $\overline{C}$ is a positive constant.

The nonlinearity in the equation is characterized by the function $f(x,u)$, which satisfies the growth conditions of the form:

$$\left(f_1\right)M_1{\left|u\right|}^{\alpha \left(x\right)-1}\le |f(x,u)|\le M_2{\left|u\right|}^{\beta \left(x\right)-1},$$

with $1<\alpha \left(x\right)\le \beta \left(x\right)<p_h^{*}\left(x\right)$, where $p_h^{*}\left(x\right)={\displaystyle \frac{N{p}_h\left(x\right)}{N-p_h\left(x\right)}}$ and $p_h\left(x\right)={\displaystyle \frac{h\left(x\right)p\left(x\right)}{h\left(x\right)+1}}$. Furthermore, the function $g\left(x\right)\in C\left(\overline{D}\right)$ is assumed to satisfy

$$\left(g\right)1\le g\left(x\right)<{\displaystyle \frac{N-\gamma \left(x\right)}{N}}{p}_h^{*}\left(x\right).$$

Remark 1. 

Consider the following assumptions and examples:

(1) 

Assume that $M_1=M_2=1$ and that $\alpha \left(x\right)=\beta \left(x\right)$. Then, the function $f(x,u)$ can be expressed as

$$f(x,u)={\left|u\right|}^{\alpha \left(x\right)-1}.$$

(2) 

A typical example of the function $g\left(x\right)$ is given by

$$g\left(x\right)=1+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{N-\gamma \left(x\right)}{N}}{p}_h^{*}\left(x\right)-1\right).$$

The primary goal of this paper is to demonstrate the existence of at least three weak solutions to the given problem under minimal assumptions on the weighted function $\kappa \left(x\right)$ and the nonlinear nonlocal term. This is achieved by employing critical point theory, developed by Bonanno and Marano [14], applied to the associated energy functional, which is derived by integrating the relevant terms of the equation over D. The use of critical point theorems enables us to establish the existence of solutions without imposing restrictive conditions on the regularity or structure of the nonlinearity. Consequently, our results offer a high degree of generality, making them applicable to a broad class of problems in mathematical physics and differential equations.

This manuscript is structured as follows: in the next section, we provide some background and auxiliary results. The final section is dedicated to presenting our main result.

2. Backgrounds and Preliminary Results

In what follows, and for $k>0$, and $p\left(x\right)\in C_{+}\left(\overline{D}\right),$ we use the following notations:

$$k^{\widehat{p}}=max\{k^{p^{-}},k^{p^{+}}\},k^{\check{p}}=min\{k^{p^{-}},k^{p^{+}}\}.$$

It is easy to verify the following properties (see [15] for further details):

(i)

$k^{{\displaystyle \frac{1}{\widehat{p}}}}=max\left\{k^{{\displaystyle \frac{1}{p^{-}}}},k^{{\displaystyle \frac{1}{p^{+}}}}\right\}$,

(ii)

$k^{{\displaystyle \frac{1}{\check{p}}}}=min\left\{k^{{\displaystyle \frac{1}{p^{-}}}},k^{{\displaystyle \frac{1}{p^{+}}}}\right\}$,

(iii)

$k^{{\displaystyle \frac{1}{\check{p}}}}=a⟺k=a^{\widehat{p}},k^{{\displaystyle \frac{1}{\widehat{p}}}}=a⟺k=a^{\check{p}}$,

(iv)

${\left(k\beta \right)}^{\check{p}}\le k^{\check{p}}{\beta }^{\check{p}}\le {\left(k\beta \right)}^{\widehat{p}}\le k^{\widehat{p}}{\beta }^{\widehat{p}}$.

Denote

$$L^{p\left(x\right)}(\kappa ,D)=\left\{u\in S\left(D\right):{\int }_D\kappa \left(x\right){\left|u\left(x\right)\right|}^{p\left(x\right)}dx<\infty \right\}$$

with a Luxemburg-type norm defined by

$${\parallel u\parallel }_{L^{p\left(x\right)}(\kappa ,D)}=inf\left\{\gamma >0:{\int }_D\kappa \left(x\right)|{\displaystyle \frac{u\left(x\right)}{\gamma }}{|}^{p\left(x\right)}dx\le 1\right\}.$$

Now, we define the variable exponent Sobolev space

$$W^{1,p\left(x\right)}\left(D\right)=\left\{u\in L^{p\left(x\right)}\left(D\right):|\nabla u|\in L^{p\left(x\right)}\left(D\right)\right\},$$

with the norm

$${\parallel u\parallel }_{W^{1,p\left(x\right)}\left(D\right)}={\parallel \nabla u\parallel }_{p\left(x\right)}+{\parallel u\parallel }_{p\left(x\right)},$$

where ${\parallel \nabla u\parallel }_{p\left(x\right)}={\parallel |\nabla u|\parallel }_{p\left(x\right)},|\nabla u|={\left({\displaystyle \sum _{i=1}^N}\right|{\displaystyle \frac{\partial u}{\partial x_i}}{|}^2)}^{{\displaystyle \frac{1}{2}}},\nabla u=\left({\displaystyle \frac{\partial u}{\partial x_1}},{\displaystyle \frac{\partial u}{\partial x_2}},...,{\displaystyle \frac{\partial u}{\partial x_N}}\right)$ is the gradient of u at $(x_1,x_2,...,x_N).$

Let

$$W^{1,p\left(x\right)}(\kappa ,D)=\{u\in L^{p\left(x\right)}\left(D\right):{\kappa }^{{\displaystyle \frac{1}{p\left(x\right)}}}|\nabla u|\in L^{p\left(x\right)}\left(D\right)\}$$

be the weighted Sobolev space, and denote by $W_0^{1,p\left(x\right)}(\kappa ,D)$ the closure of $C_0^{\infty }\left(D\right)$ in $W^{1,p\left(x\right)}(\kappa ,D)$ with the norm

$$\begin{array}{c}\parallel u\parallel =inf\left\{\gamma >0:{\int }_D\kappa \left(x\right)|{\displaystyle \frac{\nabla u\left(x\right)}{\gamma }}{|}^{p\left(x\right)}dx\le 1\right\}.\end{array}$$

Lemma 1 

([16]). If $p_1\left(x\right),p_2\left(x\right)\in C_{+}\left(\overline{D}\right)$ such that $p_1\left(x\right)\le p_2\left(x\right)$ a.e. $x\in D,$ then there is the continuous embedding $W^{1,p_2\left(x\right)}\left(D\right)\hookrightarrow W^{1,p_1\left(x\right)}\left(D\right)$.

Proposition 1 

([17]). For $p\left(x\right)\in C_{+}\left(\overline{D}\right),u\in L^{p\left(x\right)}\left(D\right),$ we have

$${\parallel u\parallel }_{p\left(x\right)}^{\check{p}}\le {\int }_D{\left|u\left(x\right)\right|}^{p\left(x\right)}dx\le {\parallel u\parallel }_{p\left(x\right)}^{\widehat{p}}.$$

Let $0<d\left(x\right)\in S\left(D\right)$, and let us define

$$L^{p\left(x\right)}(d,D):=L_{d\left(x\right)}^{p\left(x\right)}\left(D\right)=\left\{u\in S\left(D\right):{\int }_{D}d\left(x\right){\left|u\left(x\right)\right|}^{p\left(x\right)}dx<\infty \right\}$$

endowed with its Luxemburg-type norm defined as follows:

$${\parallel u\parallel }_{L_{d\left(x\right)}^{p\left(x\right)}\left(D\right)}={\parallel u\parallel }_{\left(p\right(x),d(x\left)\right)}:=inf\left\{\gamma >0:{\int }_{D}d\left(x\right)|{\displaystyle \frac{u\left(x\right)}{\gamma }}{|}^{p\left(x\right)}dx\le 1\right\}.$$

Proposition 2 

([18]). If $p\in C_{+}\left(\overline{D}\right).$ Then,

$${\parallel u\parallel }_{\left(p\right(x),d(x\left)\right)}^{\check{p}}\le {\int }_D{d\left(x\right)\left|u\left(x\right)\right|}^{p\left(x\right)}dx\le {\parallel u\parallel }_{\left(p\right(x),d(x\left)\right)}^{\widehat{p}}$$

for any $u\in L_{d\left(x\right)}^{p\left(x\right)}\left(D\right)$ and for a.e. $x\in D$.

Combining Proposition 1 with Proposition 2, one has

Lemma 2. 

Let

$${\rho }_{\kappa }\left(u\right)={\int }_D\kappa \left(x\right)|\nabla u\left(x\right){|}^{p\left(x\right)}dx.$$

For $p\in C_{+}\left(\overline{D}\right),u\in W^{1,p\left(x\right)}(\kappa ,D),$ we have

$${\parallel u\parallel }^{\check{p}}\le {\rho }_{\kappa }\left(u\right)\le {\parallel u\parallel }^{\widehat{p}}.$$

From Proposition 2.4 of [2], if $\left(w\right)$ holds, $W^{1,p\left(x\right)}(\kappa ,D)$ is a separable and reflexive Banach space.

From Theorem 2.11 of [19], if $\left(w\right)$ holds, the following embedding

$$\begin{array}{c}\hfill W^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow W^{1,p_h\left(x\right)}\left(D\right)\end{array}$$

(3)

is continuous, where

$$p_h\left(x\right)={\displaystyle \frac{p\left(x\right)h\left(x\right)}{h\left(x\right)+1}}<p\left(x\right).$$

Combining (3) with Propositions 2.7 and 2.8 in [20], we obtain that the following embedding

$$W^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow L^{\gamma \left(x\right)}\left(D\right)$$

is continuous, where

$$1\le \gamma \left(x\right)\le p_h^{*}\left(x\right)={\displaystyle \frac{N{p}_h\left(x\right)}{N-p_h\left(x\right)}}={\displaystyle \frac{Np\left(x\right)h\left(x\right)}{Nh\left(x\right)+N-p\left(x\right)h\left(x\right)}}.$$

Furthermore, the following embedding

$$W^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow \hookrightarrow L^{t\left(x\right)}\left(D\right)$$

is compact, when $1\le t\left(x\right)<p_h^{*}\left(x\right).$

In the following, we introduce a key auxiliary result essential for establishing our main theorem.

Lemma 3 

([18]). Suppose that $0\in \overline{D}$ and that the domain D has a boundary satisfying the cone property. Let $p_h\left(x\right),\gamma \left(x\right),g\left(x\right)\in C\left(\overline{D}\right)$ with $0\le \gamma \left(x\right)<N$ for all $x\in \overline{D}$. If the function g satisfies the condition

$$1\le g\left(x\right)<{\displaystyle \frac{N-\gamma \left(x\right)}{N}}{p}_h^{*}\left(x\right),\forall x\in \overline{D},$$

then the embedding

$$W^{1,p_h\left(x\right)}\left(D\right)\hookrightarrow L_{{\left|x\right|}^{-\gamma \left(x\right)}}^{g\left(x\right)}\left(D\right)$$

is compact.

Lemma 4. 

There is a constant $\tilde{c}$ such that

$${\int }_D{\displaystyle \frac{{\left|u\right|}^{g\left(x\right)}}{{\left|x\right|}^{\gamma \left(x\right)}}}dx\le \tilde{c}{(\parallel u\parallel }^{g^{+}}+{\parallel u\parallel }^{g^{-}}),\forall u\in W_0^{1,p\left(x\right)}(\kappa ,D).$$

Proof. 

By combining Proposition 2 with Lemma 3, for all $u\in W_0^{1,p\left(x\right)}(\kappa ,D)$, there is a positive constant ${\tilde{c}}_1$ such that

$$\begin{array}{cc}\hfill {\int }_D{\displaystyle \frac{{\left|u\right|}^{g\left(x\right)}}{{\left|x\right|}^{\gamma \left(x\right)}}}dx& \le {\parallel u\parallel }_{(g\left(x\right),|x{|}^{-\gamma \left(x\right)})}^{g^{+}}+{\parallel u\parallel }_{(g\left(x\right),|x{|}^{-\gamma \left(x\right)})}^{g^{-}}\hfill \\ \hfill & \le {\tilde{c}}_1\left({\parallel u\parallel }_{W^{1,p_h\left(x\right)}\left(D\right)}^{g^{+}}+{\parallel u\parallel }_{W^{1,p_h\left(x\right)}\left(D\right)}^{g^{-}}\right).\hfill \end{array}$$

Since the embedding $W_0^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow W_0^{1,p_h\left(x\right)}\left(D\right)$ holds, there is a positive constant ${\tilde{c}}_2$ such that

$${\parallel u\parallel }_{W^{1,p_h\left(x\right)}\left(D\right)}^{g^{+}}+{\parallel u\parallel }_{W^{1,p_h\left(x\right)}\left(D\right)}^{g^{-}}\le {\tilde{c}}_2{(\parallel u\parallel }^{g^{+}}+{\parallel u\parallel }^{g^{-}}).$$

Therefore, setting $\tilde{c}={\tilde{c}}_1·{\tilde{c}}_2$, we obtain

$${\int }_D{\displaystyle \frac{{\left|u\right|}^{g\left(x\right)}}{{\left|x\right|}^{\gamma \left(x\right)}}}dx\le \tilde{c}{(\parallel u\parallel }^{g^{+}}+{\parallel u\parallel }^{g^{-}}).$$

□

Define the functional ${\mathcal{I}}_{\lambda }:W_0^{1,p\left(x\right)}(\kappa ,D)\to \mathbb{R}$ by

$${\mathcal{I}}_{\lambda }\left(u\right):=\mathfrak{F}\left(u\right)-\lambda \mathfrak{G}\left(u\right),$$

where

$$\mathfrak{F}\left(u\right):={\int }_D\mathcal{B}(x,\nabla u)dx+{\displaystyle \frac{1}{q}}{\int }_D{\displaystyle \frac{{\eta \left(x\right)\left|u\right|}^q}{{\left|x\right|}^q}}dx+{\int }_D{\displaystyle \frac{{c\left(x\right)\left|u\right|}^{g\left(x\right)}}{{g\left(x\right)\left|x\right|}^{\gamma \left(x\right)}}}dx,$$

$$\mathfrak{G}\left(u\right):={\displaystyle \frac{1}{r+1}}{\left({\int }_{D}F(x,u\left(x\right))dx\right)}^{r+1},$$

with

$$F(x,u)={\int }_0^{u}f(x,\tau )d\tau ,\forall (x,u)\in D\times \mathbb{R}.$$

It is easy to see that the functionals $\mathfrak{F}$ and $\mathfrak{G}$ are continuously Gâteaux differentiable, and we have the following expressions:

$$\begin{array}{cc}\hfill {\mathfrak{F}}^{\prime }\left(u\right)\left(v\right)& ={\int }_{D}b(x,\nabla u)\nabla vdx+{\int }_D{\displaystyle \frac{{\eta \left(x\right)\left|u\right|}^{q-2}uv}{{\left|x\right|}^q}}dx+{\int }_D{\displaystyle \frac{{c\left(x\right)\left|u\right|}^{g\left(x\right)-2}uv}{{\left|x\right|}^{\gamma \left(x\right)}}}dx,\hfill \end{array}$$

and

$${\mathfrak{G}}^{\prime }\left(u\right)\left(v\right)={\left({\int }_{D}F(x,u)dx\right)}^r{\int }_{D}f(x,u)vdx,\forall u,v\in W_0^{1,p\left(x\right)}(\kappa ,D).$$

A function $u\in W_0^{1,p\left(x\right)}(\kappa ,D)$ is called a weak solution of problem $\left(1\right)$ if it satisfies the variational equation

$${\mathcal{I}}_{\lambda }^{\prime }\left(u\right)\left(w\right)={\mathfrak{F}}^{\prime }\left(u\right)\left(w\right)-\lambda {\mathfrak{G}}^{\prime }\left(u\right)\left(w\right)=0,\forall w\in W_0^{1,p\left(x\right)}(\kappa ,D).$$

Lemma 5. 

The functional ${\mathfrak{F}}^{\prime }$ is coercive and strictly monotone in $W_0^{1,p\left(x\right)}(\kappa ,D).$

Proof. 

For every $u\in W_0^{1,p\left(x\right)}(\kappa ,D)\setminus \left\{0\right\}$, using (2), we obtain

$$\begin{array}{cc}\hfill {\mathfrak{F}}^{\prime }\left(u\right)\left(u\right)& ={\int }_{D}b(x,\nabla u)·\nabla udx+{\int }_D{\displaystyle \frac{{\eta \left(x\right)\left|u\right|}^{q-2}{u}^2}{{\left|x\right|}^q}}dx+{\int }_D{\displaystyle \frac{{c\left(x\right)\left|u\right|}^{g\left(x\right)-2}{u}^2}{{\left|x\right|}^{\gamma \left(x\right)}}}dx\hfill \\ & \ge {\mathcal{C}}_2{\int }_D\kappa \left(x\right){|\nabla u|}^{p\left(x\right)}dx\hfill \\ & \ge {\mathcal{C}}_2{min\{\parallel u\parallel }^{p^{+}}{,\parallel u\parallel }^{p^{-}}\}.\hfill \end{array}$$

Thus, we obtain

$$\underset{\parallel u\parallel \to \infty }{lim}{\displaystyle \frac{{\mathfrak{F}}^{\prime }\left(u\right)\left(u\right)}{\parallel u\parallel }}\ge {\mathcal{C}}_2\underset{\parallel u\parallel \to \infty }{lim}{\displaystyle \frac{min\{\parallel u{\parallel }^{p^{+}},\parallel u{\parallel }^{p^{-}}\}}{\parallel u\parallel }}=+\infty .$$

Consequently, ${\mathfrak{F}}^{\prime }$ is coercive due to the continuity of $p\left(x\right)$ over $\overline{D}$.

According to Equation (4) in [21], for all $x,y\in {\mathbb{R}}^N$, there is a positive constant $C_p$ such that if $p\ge 2$, than

$${\langle \left|x\right|}^{p-2}x-{\left|y\right|}^{p-2}y,x-y\rangle \ge C_p{|x-y|}^p,$$

moreover, for $1<p<2,\mathrm{and}(x,y)\ne (0,0)$, one has

$${\langle \left|x\right|}^{p-2}x-{\left|y\right|}^{p-2}y,x-y\rangle \ge {\displaystyle \frac{C_p{|x-y|}^2}{\left(\right|x|+{\left|y\right|)}^{2-p}}},$$

here, $\langle ·,·\rangle$ denotes the standard inner product in ${\mathbb{R}}^N$.

Thus, for any $u,v$ that belong in $X:=W_0^{1,p\left(x\right)}(\kappa ,D)$, with $u\ne v$, we obtain

$$\begin{array}{cc}\hfill \langle {\mathfrak{F}}^{\prime }\left(u\right)-{\mathfrak{F}}^{\prime }\left(v\right),u-v\rangle & \ge {\int }_D{\kappa \left(x\right)\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u-{|\nabla v|}^{p\left(x\right)-2}\nabla v)·(\nabla u-\nabla v)dx\hfill \\ & +{\int }_D{\displaystyle \frac{\eta \left(x\right)}{{\left|x\right|}^q}}{\left(\right|u|}^{q-2}u-{\left|v\right|}^{q-2}v)(u-v)dx\hfill \\ & +{\int }_D{\displaystyle \frac{c\left(x\right)}{{\left|x\right|}^{\gamma \left(x\right)}}}{\left(\right|u|}^{g\left(x\right)-2}u-{\left|v\right|}^{g\left(x\right)-2}v)(u-v)dx\hfill \\ & >0.\hfill \end{array}$$

Hence, ${\mathfrak{F}}^{\prime }$ is strictly monotone in $W_0^{1,p\left(x\right)}(\kappa ,D)$. □

Lemma 6. 

The functional ${\mathfrak{F}}^{\prime }$ is a mapping of $\left(S_{+}\right)$-type, i.e., if $u_n\rightharpoonup u$ in $W_0^{1,p\left(x\right)}(\kappa ,D),$ and ${{\displaystyle \overline{lim}}}_{n\to \infty }\langle {\mathfrak{F}}^{\prime }\left(u_n\right)-{\mathfrak{F}}^{\prime }\left(u\right),u_n-u)\rangle \le 0,$ then $u_n\to u$ in $W_0^{1,p\left(x\right)}(\kappa ,D).$

Proof. 

Let $u_n\rightharpoonup u$ in $W_0^{1,p\left(x\right)}(\kappa ,D),$ and ${{\displaystyle \overline{lim}}}_{n\to \infty }\langle {\mathfrak{F}}^{\prime }\left(u_n\right)-{\mathfrak{F}}^{\prime }\left(u\right),u_n-u\rangle \le 0.$

Due to the strict monotonicity of ${\mathfrak{F}}^{\prime }$ in $W_0^{1,p\left(x\right)}(\kappa ,D),$ one has

$$\underset{n\to \infty }{lim}\langle {\mathfrak{F}}^{\prime }\left(u_n\right)-{\mathfrak{F}}^{\prime }\left(u\right),u_n-u\rangle =0,$$

while

$$\begin{array}{cc}\hfill \langle {\mathfrak{F}}^{\prime }\left(u_n\right)-{\mathfrak{F}}^{\prime }\left(u\right),u_n-u\rangle & \ge {\int }_D\kappa \left(x\right)\left(\right|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u)(\nabla u_n-\nabla u)dx\hfill \\ & +{\int }_D\left({\displaystyle \frac{\eta \left(x\right)|u_n{|}^{q-2}}{{\left|x\right|}^q}}{u}_n(u_n-u)-{\displaystyle \frac{{\eta \left(x\right)\left|u\right|}^{q-2}}{{\left|x\right|}^q}}u(u_n-u)\right)dx\hfill \\ & +{\int }_D\left({\displaystyle \frac{c\left(x\right)|u_n{|}^{g\left(x\right)-2}}{{\left|x\right|}^{\gamma \left(x\right)}}}{u}_n(u_n-u)-{\displaystyle \frac{{c\left(x\right)\left|u\right|}^{g\left(x\right)-2}}{{\left|x\right|}^{\gamma \left(x\right)}}}u(u_n-u)\right)dx,\hfill \end{array}$$

Further, by (2) one has

$${\overline{lim}}_{n\to \infty }{\int }_D\kappa \left(x\right)(|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u)(\nabla u_n-\nabla u)dx\le 0,$$

then $u_n\to u$ in $W_0^{1,p\left(x\right)}(\kappa ,D)$ via Lemma 3.2 in [22]. □

Lemma 7. 

The functional ${\mathfrak{F}}^{\prime }$ is a homeomorphism.

Proof. 

Since ${\mathfrak{F}}^{\prime }$ is strictly monotone, this implies its injectivity. Moreover, since ${\mathfrak{F}}^{\prime }$ is coercive, it follows that ${\mathfrak{F}}^{\prime }$ is also surjective. Consequently, ${\mathfrak{F}}^{\prime }$ has a well-defined inverse mapping.

Next, we establish the continuity of the inverse mapping ${\left({\mathfrak{F}}^{\prime }\right)}^{-1}$.

Let ${\tilde{f}}_n,\tilde{f}\in {\left(W_0^{1,p\left(x\right)}(\kappa ,D)\right)}^{*}$ such that ${\tilde{f}}_n\to \tilde{f}$. Our goal is to prove that

$${\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left({\tilde{f}}_n\right)\to {\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left(\tilde{f}\right).$$

Define $u_n={\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left({\tilde{f}}_n\right)$ and $u={\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left(\tilde{f}\right)$, which implies

$${\mathfrak{F}}^{\prime }\left(u_n\right)={\tilde{f}}_n\mathrm{and}{\mathfrak{F}}^{\prime }\left(u\right)=\tilde{f}.$$

By the coercivity of ${\mathfrak{F}}^{\prime }$, the sequence $u_n$ is bounded in $W_0^{1,p\left(x\right)}(\kappa ,D)$. Without a loss of generality, assume $u_n\rightharpoonup u_0$. Then, we obtain

$$\underset{n\to \infty }{lim}\langle {\mathfrak{F}}^{\prime }\left(u_n\right)-{\mathfrak{F}}^{\prime }\left(u\right),u_n-u_0\rangle =\underset{n\to \infty }{lim}\langle {\tilde{f}}_n-\tilde{f},u_n-u_0\rangle =0.$$

Since ${\mathfrak{F}}^{\prime }$ is of $\left(S_{+}\right)$-type, it follows that $u_n\to u_0$ in $W_0^{1,p\left(x\right)}(\kappa ,D)$, which ensures that ${\mathfrak{F}}^{\prime }\left(u_n\right)\to {\mathfrak{F}}^{\prime }\left(u_0\right)$. Given that ${\mathfrak{F}}^{\prime }\left(u_n\right)\to {\mathfrak{F}}^{\prime }\left(u\right)$, we conclude that ${\mathfrak{F}}^{\prime }\left(u\right)={\mathfrak{F}}^{\prime }\left(u_0\right)$. Since ${\mathfrak{F}}^{\prime }$ is injective, it follows that $u=u_0$, and hence $u_n\to u$.

Thus, we have shown that ${\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left({\tilde{f}}_n\right)\to {\left({\mathfrak{F}}^{\prime }\right)}^{-1}\left(\tilde{f}\right)$, proving the continuity of ${\left({\mathfrak{F}}^{\prime }\right)}^{-1}$. □

Lemma 8 

(Hölder type inequality [20,23]). Let $p\left(x\right),q\left(x\right),s\left(x\right)\ge 1$ be three measurable functions on D satisfying

$${\displaystyle \frac{1}{s\left(x\right)}}={\displaystyle \frac{1}{p\left(x\right)}}+{\displaystyle \frac{1}{q\left(x\right)}},\mathit{for}\mathit{almost}\mathit{every}x\in D.$$

Suppose that $f\in L^{p\left(x\right)}\left(D\right)$ and $g\in L^{q\left(x\right)}\left(D\right)$. Then, the product $fg$ belongs to $L^{s\left(x\right)}\left(D\right)$ and satisfies the following inequality:

$${\parallel fg\parallel }_{s\left(x\right)}\le {2\parallel f\parallel }_{p\left(x\right)}{\parallel g\parallel }_{q\left(x\right)}.$$

(4)

Lemma 9. 

The functional ${\mathfrak{G}}^{\prime }:X:=W_0^{1,p\left(x\right)}(\kappa ,D)\to {\left(W_0^{1,p\left(x\right)}(\kappa ,D)\right)}^{*}$ is compact.

Proof. 

The condition $\left(f_1\right)$ and the compact embedding $W_0^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow \hookrightarrow L^{\beta \left(x\right)}\left(D\right)$, where $1\le \beta \left(x\right)<p_h^{*}\left(x\right)$, ensure the compactness of ${\mathfrak{G}}^{\prime }\left(u\right)$.

Consider a sequence ${\left(u_n\right)}_n\subset X$ such that $u_n\rightharpoonup u$.

Due to the compact embedding $W_0^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow \hookrightarrow L^{\beta \left(x\right)}\left(D\right)$, there is a subsequence, still denoted by ${\left(u_n\right)}_n$, such that $u_n\to u$ strongly in $L^{\beta \left(x\right)}\left(D\right)$ and $u_n\left(x\right)\to u\left(x\right)$ almost everywhere in D.

Since $F(x,u)$ is continuous with respect to u, it follows that

$$F(x,u_n)\to F(x,u)\mathrm{almost}\mathrm{everywhere}\mathrm{in}D.$$

Furthermore, there is a constant $C>0$ such that

$$|F(x,u_n)|\le C|u_n{|}^{\beta \left(x\right)}.$$

By applying the Dominated Convergence Theorem, we conclude that

$${\int }_{D}F(x,u_n)dx\to {\int }_{D}F(x,u)dx\mathrm{as}n\to +\infty .$$

(5)

From condition $\left(f_1\right)$, it follows that the Nemytskii operator $N_f$ given by

$$N_f\left(u\right)\left(x\right)=f(x,u\left(x\right))$$

is continuous, as $f:D\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying $\left(f_1\right)$. Consequently, we obtain

$$N_f\left(u_n\right)\to N_f\left(u\right)\mathrm{in}{L}^{{\displaystyle \frac{\beta \left(x\right)}{\beta \left(x\right)-1}}}\left(D\right).$$

Applying Hölder’s inequality, for every $v\in W_0^{1,p\left(x\right)}(\kappa ,D)$, we derive

$$\begin{array}{cc}\hfill \left|{\int }_{D}f(x,u_n)vdx-{\int }_{D}f(x,u)vdx\right|& \le {\int }_D\left|(f(x,u_n)-f(x,u))v\right|dx\hfill \\ \hfill & \le 2\parallel N_f\left(u_n\right)-N_f\left(u\right){\parallel }_{{\displaystyle \frac{\beta \left(x\right)}{\beta \left(x\right)-1}}}{\parallel v\parallel }_{\beta \left(x\right)}\hfill \\ \hfill & \le 2c_{\beta }\parallel N_f\left(u_n\right)-N_f\left(u\right){\parallel }_{{\displaystyle \frac{\beta \left(x\right)}{\beta \left(x\right)-1}}}\parallel v\parallel ,\hfill \end{array}$$

where $c_{\beta }$ is the embedding constant of

$$W_0^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow L^{\beta \left(x\right)}\left(D\right),1\le \beta \left(x\right)<p_h^{*}\left(x\right).$$

Thus, from (5) and the above inequality, we conclude that ${\mathfrak{G}}^{\prime }\left(u_n\right)\to {\mathfrak{G}}^{\prime }\left(u\right)$ in $X^{*}$, proving that ${\mathfrak{G}}^{\prime }$ is completely continuous and, consequently, compact. □

In what follows, we announce the critical point theorem, which constitutes our principal tool to obtain our result.

Theorem 1 

([14], Theorem 3.6). Let Y be a reflexive Banach space over $\mathbb{R}$, and let $\mathfrak{F}:Y\to \mathbb{R}$ be a coercive functional that is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous. Suppose that the Gâteaux derivative of $\mathfrak{F}$ has a continuous inverse on the dual space $Y^{*}$. Additionally, let $\mathfrak{G}:Y\to \mathbb{R}$ be another continuously Gâteaux differentiable functional whose derivative is compact. Assume the following conditions hold:

$$\left(a_0\right)\underset{Y}{inf}\mathfrak{F}=\mathfrak{F}\left(0\right)=\mathfrak{G}\left(0\right)=0.$$

There is a constant $d>0$ and a point $\overline{v}\in Y$ such that $\mathfrak{F}\left(\overline{v}\right)>d$, and the following conditions are satisfied:

$$\left(a_1\right){\displaystyle \frac{{sup}_{\mathfrak{F}\left(v\right)<d}\mathfrak{G}\left(v\right)}{d}}<{\displaystyle \frac{\mathfrak{G}\left(\overline{v}\right)}{\mathfrak{F}\left(\overline{v}\right)}}.$$

$$\left(a_2\right)\mathit{For}\mathit{each}\lambda \in {\mathsf{\Lambda }}_d:=\left({\displaystyle \frac{\mathfrak{F}\left(\overline{v}\right)}{\mathfrak{G}\left(\overline{v}\right)}},{\displaystyle \frac{d}{{sup}_{\mathfrak{F}\left(v\right)\le d}\mathfrak{G}\left(v\right)}}\right),\mathit{the}\mathit{functional}{I}_{\lambda }:=\mathfrak{F}-\lambda \mathfrak{G}\mathit{is}\mathit{coercive}.$$

Under these conditions, for every $\lambda \in {\mathsf{\Lambda }}_d$, the functional $\mathfrak{F}-\lambda \mathfrak{G}$ possesses at least three distinct critical points in Y.

3. Main Results

In this section, we establish a theorem guaranteeing our multiplicity result.

To proceed, we recall the Hardy inequality (see Lemma 2.1 in [1] for further details), which states that for $1<p<N$, the following holds:

$${\int }_D{\displaystyle \frac{{\left|u\left(x\right)\right|}^p}{{\left|x\right|}^p}}dx\le {\displaystyle \frac{1}{H}}{\int }_D{|\nabla u|}^{p}dx,\forall u\in W_0^{1,p}\left(D\right),$$

(6)

where $H={\left({\displaystyle \frac{N-p}{p}}\right)}^p$ is the optimal constant.

Using this inequality together with Lemma 1 and the assumption that $1<q<p_h\left(x\right)<N$, we obtain the continuous embeddings

$$W_0^{1,p\left(x\right)}(\kappa ,D)\hookrightarrow W_0^{1,p_h\left(x\right)}\left(D\right)\hookrightarrow W_0^{1,q}\left(D\right),$$

(7)

which leads to the inequality

$${\int }_D{\displaystyle \frac{{\left|u\left(x\right)\right|}^q}{{\left|x\right|}^q}}dx\le {\displaystyle \frac{1}{H}}{\int }_D{|\nabla u|}^{q}dx,\forall u\in W_0^{1,p\left(x\right)}(\kappa ,D),$$

(8)

where $H={\left({\displaystyle \frac{N-q}{q}}\right)}^q$.

With these preliminaries in place, we now present our main result. To this end, we define

$$\tilde{R}\left(x\right):=sup\left\{\tilde{R}>0\mid B(x,\tilde{R})\subseteq D\right\}$$

for each $x\in D$, where $B(x,\tilde{R})$ denotes a ball centered at x with radius $\tilde{R}$. It is evident that there is a point $x^0\in D$ such that $B(x^0,R)\subseteq D$, where

$$R=\underset{x\in D}{sup}\tilde{R}\left(x\right).$$

In the sequel, we introduce the constant m, defined as

$$m=max\left\{{\tilde{C}}_1{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)},{\tilde{C}}_1{\parallel \tilde{\nu }\left(x\right)\parallel }_{L^{p^{\prime }\left(x\right)}\left(D\right)},{\displaystyle \frac{{\parallel \eta \parallel }_{\infty }}{qH}},{\displaystyle \frac{\tilde{c}{\parallel C\parallel }_{\infty }}{g^{-}}}\left({\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}^{{\displaystyle \frac{g^{+}}{\widehat{p}}}}+{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}^{{\displaystyle \frac{g^{-}}{\widehat{p}}}}\right)\right\},$$

where

$$\mathfrak{B}:=\overline{B}(x^0,R)\setminus B(x^0,R/2).$$

Theorem 2. 

Suppose that $p^{-}>{\beta }^{+}(r+1)$, and that there are two positive constants d and δ satisfying

$${\displaystyle \frac{1}{p^{+}}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\check{p}}{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}=d.$$

Define the constants $A_{\delta }$ and $B_d$, where $A_{\delta }<B_d$, as follows:

$$A_{\delta }:={\displaystyle \frac{m\left[{\left(\frac{2\delta }{R}\right)}^{\widehat{p}}+{\left(\frac{2\delta }{R}\right)}^q\left|\mathfrak{B}\right|+\frac{2\delta }{R}{\left|\mathfrak{B}\right|}^{\frac{1}{\widehat{p}}}+\frac{2\delta }{R}\right]}{\frac{M_1^{r+1}}{(r+1){\left({\alpha }^{+}\right)}^{r+1}}{\delta }^{\check{\alpha }(r+1)}{\left|B(x^0,R/2)\right|}^{r+1}}},$$

$$B_d:={\displaystyle \frac{d}{\frac{M_2^{r+1}{c}_{\beta }^{\widehat{\beta }(r+1)}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}{\left[{\left(\frac{p^{+}}{\overline{C}}d\right)}^{\frac{1}{\check{p}}}\right]}^{\widehat{\beta }(r+1)}}}.$$

Then, for any $\lambda \in (A_{\delta },B_d)$, problem (1) has at least three weak solutions.

Proof. 

It is important to note that $\mathfrak{F}$ and $\mathfrak{G}$ satisfy the regularity assumptions stated in Theorem 1. We now proceed to verify the fulfillment of conditions $\left(a_1\right)$ and $\left(a_2\right)$. To this end, we define

$${\displaystyle \frac{1}{p^{+}}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\check{p}}{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}=d$$

and consider a function $v_{\delta }\in X$ given by

$$v_{\delta }\left(x\right):=\left\{\begin{array}{cc}0,\hfill & x\in D\setminus B(x^0,R),\hfill \\ {\displaystyle \frac{2\delta }{R}}\left(R-|x-x^0|\right),\hfill & x\in \mathfrak{B}:=\overline{B}(x^0,R)\setminus B(x^0,R/2),\hfill \\ \delta ,\hfill & x\in \overline{B}(x^0,R/2).\hfill \end{array}\right.$$

Then, applying condition $\left(\tilde{B}2\right)$, the definition of $\mathfrak{F}$, and Lemma 4, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{p^{+}}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\check{p}}{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}& <\overline{C}{\int }_D{\displaystyle \frac{\kappa \left(x\right)}{p\left(x\right)}}{|\nabla v_{\delta }|}^{p\left(x\right)}dx\hfill \\ \hfill & <\mathfrak{F}\left(v_{\delta }\right)\hfill \\ \hfill & \le {\int }_D{\tilde{C}}_1\left[\tilde{\nu }\left(x\right)|\nabla v_{\delta }|+\kappa \left(x\right)|\nabla v_{\delta }{|}^{p\left(x\right)}\right]dx+{\displaystyle \frac{{\parallel \eta \parallel }_{\infty }}{q}}{\int }_D{\displaystyle \frac{|v_{\delta }{|}^q}{{\left|x\right|}^q}}dx+{\displaystyle \frac{{\parallel C\parallel }_{\infty }}{g^{-}}}{\int }_D{\displaystyle \frac{|v_{\delta }{|}^{g\left(x\right)}}{{\left|x\right|}^{\gamma \left(x\right)}}}dx\hfill \\ \hfill & \le {\tilde{C}}_1{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\widehat{p}}{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}+{\tilde{C}}_1\parallel \tilde{\nu }{\left(x\right)\parallel }_{L^{p^{\prime }\left(x\right)}\left(D\right)}{\parallel \nabla v_{\delta }\parallel }_{L^{p\left(x\right)}\left(D\right)}\hfill \\ \hfill & +{\displaystyle \frac{{\parallel \eta \parallel }_{\infty }}{qH}}{\int }_D{|\nabla v_{\delta }|}^{q}dx+{\displaystyle \frac{\tilde{c}{\parallel C\parallel }_{\infty }}{g^{-}}}\left(\parallel v_{\delta }{\parallel }^{g^{+}}+{\parallel v_{\delta }\parallel }^{g^{-}}\right)\hfill \\ \hfill & \le {\tilde{C}}_1{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\widehat{p}}{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}+{\tilde{C}}_1\parallel \tilde{\nu }{\left(x\right)\parallel }_{L^{p^{\prime }\left(x\right)}\left(D\right)}\left({\displaystyle \frac{2\delta }{R}}\right){\left|\mathfrak{B}\right|}^{{\displaystyle \frac{1}{\widehat{p}}}}\hfill \\ \hfill & +{\displaystyle \frac{{\parallel \eta \parallel }_{\infty }}{qH}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^q\left|\mathfrak{B}\right|+{\displaystyle \frac{\tilde{c}{\parallel C\parallel }_{\infty }}{g^{-}}}{\displaystyle \frac{2\delta }{R}}\left({\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}^{{\displaystyle \frac{g^{+}}{\widehat{p}}}}+{\parallel \kappa \parallel }_{L^1\left(\mathfrak{B}\right)}^{{\displaystyle \frac{g^{-}}{\widehat{p}}}}\right)\hfill \end{array}$$

$$\hspace{1em}\hspace{1em}\le m\left[{\left({\displaystyle \frac{2\delta }{R}}\right)}^{\widehat{p}}+{\left({\displaystyle \frac{2\delta }{R}}\right)}^q\left|\mathfrak{B}\right|+{\displaystyle \frac{2\delta }{R}}{\left|\mathfrak{B}\right|}^{{\displaystyle \frac{1}{\widehat{p}}}}+{\displaystyle \frac{2\delta }{R}}\right].$$

(9)

Therefore, we obtain $\mathfrak{F}\left(v_{\delta }\right)>d$. Moreover, one has

$$\begin{array}{ccc}\hfill \mathfrak{G}\left(v_{\delta }\right)& \ge & {\displaystyle \frac{1}{r+1}}{\left({\int }_{D}F(x,v_{\delta })dx\right)}^{r+1}\hfill \\ & \ge & {\displaystyle \frac{M_1^{r+1}}{(r+1){\left({\alpha }^{+}\right)}^{r+1}}}{\left({\int }_{B(x^0,R/2)}{\left|\delta \right|}^{\alpha \left(x\right)}dx\right)}^{r+1}\hfill \\ & \ge & {\displaystyle \frac{M_1^{r+1}}{(r+1){\left({\alpha }^{+}\right)}^{r+1}}}{\delta }^{\check{\alpha }(r+1)}{\left|B(x^0,R/2)\right|}^{r+1}.\hfill \end{array}$$

(10)

Additionally, for every $u\in {\mathfrak{F}}^{-1}\left(\right]-\infty ,d\left]\right)$, the following holds:

$${\displaystyle \frac{\overline{C}}{p^{+}}}{\parallel u\parallel }^{\check{p}}\le d.$$

(11)

As a result, we deduce that

$$\parallel u\parallel \le {\left({\displaystyle \frac{p^{+}}{\overline{C}}}\mathfrak{F}\left(u\right)\right)}^{{\displaystyle \frac{1}{\check{p}}}}<{\left({\displaystyle \frac{p^{+}}{\overline{C}}}d\right)}^{{\displaystyle \frac{1}{\check{p}}}}.$$

Moreover, under assumption $\left(f_1\right)$, we obtain the following upper bound:

$$\begin{array}{ccc}\hfill \mathfrak{G}\left(u\right)& \le & {\displaystyle \frac{1}{r+1}}{\left({\int }_{D}F(x,u\left(x\right))dx\right)}^{r+1}\hfill \\ & \le & {\displaystyle \frac{M_2^{r+1}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}}{\left({\int }_D{\left|u\right|}^{\beta \left(x\right)}dx\right)}^{r+1}\hfill \\ & \le & {\displaystyle \frac{M_2^{r+1}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}}{\left({\parallel u\parallel }_{\beta }^{\widehat{\beta }}\right)}^{r+1}\hfill \\ & \le & {\displaystyle \frac{M_2^{r+1}{c}_{\beta }^{\widehat{\beta }(r+1)}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}}{\parallel u\parallel }^{\widehat{\beta }(r+1)}.\hfill \end{array}$$

(12)

As a consequence, we obtain the following result:

$$\begin{array}{ccc}\hfill \underset{\mathfrak{F}\left(u\right)<d}{sup}\mathfrak{G}\left(u\right)& \le & {\displaystyle \frac{M_2^{r+1}{c}_{\beta }^{\widehat{\beta }(r+1)}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}}{\left[{\left({\displaystyle \frac{p^{+}}{\overline{C}}}d\right)}^{{\displaystyle \frac{1}{\check{p}}}}\right]}^{\widehat{\beta }(r+1)}.\hfill \end{array}$$

Moreover, it follows that

$${\displaystyle \frac{1}{d}}\underset{\mathfrak{F}\left(u\right)<d}{sup}\mathfrak{G}\left(u\right)<{\displaystyle \frac{1}{\lambda }}.$$

Furthermore, by utilizing inequality (12) once again, we establish the coercivity of ${\mathcal{I}}_{\lambda }$ for every $\lambda >0$. Specifically, for sufficiently large $\parallel u\parallel$, we derive the following estimate:

$$\mathfrak{F}\left(u\right)-\lambda \mathfrak{G}\left(u\right)\ge {\displaystyle \frac{\overline{C}}{p^{+}}}{\parallel u\parallel }^{p^{-}}-\lambda {\displaystyle \frac{M_2^{r+1}{c}_{\beta }^{\widehat{\beta }(r+1)}}{(r+1){\left({\beta }^{-}\right)}^{r+1}}}{\left({\parallel u\parallel }^{\widehat{\beta }}\right)}^{r+1}.$$

Since $p^{-}>{\beta }^{+}(r+1)$, we conclude that the desired result holds.

Finally, considering the inclusion

$${\overline{\mathsf{\Lambda }}}_d:=\left(A_{\delta },B_d\right)\subseteq \left({\displaystyle \frac{\mathfrak{F}\left(v_{\delta }\right)}{\mathfrak{G}\left(v_{\delta }\right)}},{\displaystyle \frac{d}{{sup}_{\mathfrak{F}\left(u\right)<d}\mathfrak{G}\left(u\right)}}\right),$$

we apply Theorem 1 to deduce that for each $\lambda \in {\overline{\mathsf{\Lambda }}}_d$, the functional $\mathfrak{F}-\lambda \mathfrak{G}$ admits at least three critical points in $W_0^{1,p\left(x\right)}(\kappa ,D)$ that correspond to weak solutions of problem (1). □

4. Conclusions

In this study, we have analyzed a class of degenerate Leray–Lions-type elliptic equations involving a double Hardy potential. Under appropriate conditions on the domain and the problem’s coefficients, we established the existence and quantitative properties of weak solutions. Specifically, assuming that the domain satisfies the cone property and that the coefficients exhibit controlled growth, we proved the existence of three weak solutions. Additionally, a perspective of our work is to study our problem with a nonlinearity of the form:

$${f\left(x\right)\left|u\right|}^{s\left(x\right)-2}u{\left({\int }_{\mathcal{D}}{\displaystyle \frac{f\left(x\right)}{s\left(x\right)}}{\left|u\right|}^{s\left(x\right)}dx\right)}^r,$$

where the weight function f is an element of the variable Lebesgue space $L^{\gamma \left(x\right)}\left(\mathcal{D}\right)$, assumed to be positive and possibly singular in $\mathcal{D}$ in the superlinear case, while it may change sign in the sublinear case.