Three Solutions for a Double-Phase Variable-Exponent Kirchhoff Problem

Abstract

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an n-dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as $C^1$-smoothness, the $\left(S_{+}\right)$-condition, and sequential weak lower semicontinuity.

1. Introduction

In this article, we study the following double-phase variable-exponent Kirchhoff problem:

$$\left\{\begin{array}{cc}{\displaystyle -\mathcal{M}\left({\int }_{\Omega }\frac{{|\nabla u|}^{p\left(x\right)}}{p\left(x\right)}+\mu \left(x\right)\frac{{|\nabla u|}^{q\left(x\right)}}{q\left(x\right)}\right)}\hfill & \\ \times \mathrm{div}\left({|\nabla u|}^{p\left(x\right)-2}\nabla u+\mu \left(x\right){|\nabla u|}^{q\left(x\right)-2}\nabla u\right)=\lambda f(x,u)\mathrm{in}\Omega ,\hfill \\ u=0\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

($𝒫$)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$$(N\ge 2)$ with a Lipschitz boundary; $p,q\in C_{+}\left(\overline{\Omega }\right)$; f is a Carathéodory function; and $\lambda >0$ is a real parameter.

The problem ($𝒫$) indicates a generalization of the Kirchhoff equation [1]. Initially, Kirchhoff suggested a model given by the equation

$$\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{P_0}{h}}+{\displaystyle \frac{E}{2l}}{\int }_0^l{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^{2}dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,$$

where $\rho$, $P_0$, h, E, l are constants. It extends the classical D’Alambert wave equation by considering the effects of changes in the lengths of the strings during vibrations. However, since then, this model and its various perturbed versions, especially in the variable exponent setting, have been studied intensively by many authors.

Equations of the form ($𝒫$) appear in models involving materials with nonuniform (anisotropic) properties, fluid mechanics, image processing, and elasticity in nonhomogeneous materials. Consequently, problem ($𝒫$) can be used to model various real-world phenomena, primarily due to the presence of the operator

$$\mathrm{div}\left({|\nabla u|}^{p\left(x\right)-2}\nabla u+\mu \left(x\right){|\nabla u|}^{q\left(x\right)-2}\nabla u\right)$$

(1)

which governs anisotropic and heterogeneous diffusion associated with the energy functional

$$u\to {\int }_{\Omega }\left({\displaystyle \frac{{|\nabla u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{{|\nabla u|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx,u\in W_0^{1,\mathcal{H}}(\Omega )$$

(2)

which is called a "double-phase” operator. This functional displays varying ellipticity depending on the regions where the weight function $\mu (·)$ vanishes, thereby transitioning between two distinct elliptic phases. Zhikov [2] was the first to study this type of functional with constant exponents, aiming to describe the behavior of strongly anisotropic materials. In the framework of elasticity theory, the function $\mu (·)$ reflects the geometric structure of composites made from two distinct materials characterized by power-law hardening exponents $p(·)$ and $q(·)$. Since then, numerous studies have explored this topic due to its wide applicability across various disciplines (e.g., [3,4,5,6,7,8,9,10,11,12,13]). The interested reader may also refer to [14] for an overview of the isotropic and anisotropic double-phase problems.

The analysis of problem ($𝒫$) requires the framework of Musielak–Orlicz Sobolev spaces. In the context of fluid mechanics, Orlicz spaces provide a natural and more flexible framework than classical $L^p$ spaces for the modeling of materials whose rheological behavior deviates from standard power-law profiles. They accommodate more general growth conditions such as logarithmic corrections $t^{p}log(1+t)$ or exponential-type responses that frequently arise in the study of slow or fast diffusion, polymeric flows, and plasticity. These phenomena are often encountered in non-Newtonian and complex fluids, where the stress–strain relationship is nonlinear and varies with physical parameters like pressure, electric field, or temperature.

We refer to the following related papers where double-phase variable-exponent Kirchhoff problems are studied. In [15], the authors study a class of double-phase variable-exponent problems of the Kirchhoff type. Using the sub-supersolution method within an appropriate Musielak–Orlicz Sobolev space framework, they demonstrate the existence of at least one nonnegative solution. Moreover, by imposing an additional assumption on the nonlinearity, the authors employ variational arguments to establish the existence of a second nonnegative solution. In [16], the authors address a class of Kirchhoff-type problems in a double-phase setting with a small perturbation. The authors provide a new, less restrictive assumption than the (AR)-condition, which is a crucial tool in applying the Mountain Pass Theorem, under which the problem admits at least two weak solutions. The proof is based on variational arguments, utilizing the Mountain Pass Theorem with the Cerami condition. In contrast to the works [15,16], the principal novelty of this paper lies in the introduction of a new n-dimensional vector inequality, presented in Proposition 9. This inequality serves as a crucial auxiliary tool for establishing key regularity properties of the energy functional $\mathcal{K}$ and its derivative ${\mathcal{K}}^{\prime }$, including $C^1$-smoothness, the $\left(S_{+}\right)$-condition, and sequential weak lower semicontinuity.

The paper is organized as follows. In Section 2, we first provide some background for the theory of variable Sobolev spaces $W_0^{1,p\left(x\right)}(\Omega )$ and the Musielak–Orlicz Sobolev space $W_0^{1,\mathcal{H}}(\Omega )$. In this section, we prove Proposition 9, which is the main originality of the paper, as well as the crucial auxiliary result showing that the functional $\mathcal{K}$ is continuously Gâteaux-differentiable, which is one of the main difficulties in the study of problem ($𝒫$). In Section 3, we obtain another crucial auxiliary result, namely Lemmas 2 and 3, where we show the required regularity assumptions of the corresponding functionals ${\mathcal{I}}_{\lambda }$, $\mathcal{K}$, and $\mathcal{J}$ of problem ($𝒫$). Then, we show that problem ($𝒫$) admits at least three distinct weak solutions by applying the well-known critical point result given by Bonanno and and Marano [17] (Theorem 3.6).

2. Mathematical Background and Auxiliary Results

We start with some basic concepts of variable Lebesgue–Sobolev spaces. For more details, and for the proofs of the following propositions, we refer the reader to [18,19,20,21,22].

Define the set

$$C_{+}\left(\overline{\Omega }\right)=\left\{h\in C\left(\overline{\Omega }\right):h\left(x\right)>1\mathrm{for}\mathrm{all}x\in \overline{\Omega }\right\}.$$

For $h\in C_{+}\left(\overline{\Omega }\right)$, we denote

$$h^{-}:=\underset{x\in \overline{\Omega }}{min}h\left(x\right)\le h\left(x\right)\le h^{+}:=\underset{x\in \overline{\Omega }}{max}h\left(x\right)<\infty .$$

We also use the following notations ($a,b,t\in {\mathbb{R}}_{+}$):

$$min{t}^{h\left(x\right)}=t^{h_{-}}:=\left\{\begin{array}{cc}{t}^{h^{+}},\hfill & t<1\hfill \\ t^{h^{-}},\hfill & t\ge 1\hfill \end{array}\right.;max{t}^{h\left(x\right)}=t^{h_{+}}=:\left\{\begin{array}{cc}{t}^{h^{-}},\hfill & t<1\hfill \\ t^{h^{+}},\hfill & t\ge 1\hfill \end{array}\right.$$

and

$$t^{a\vee b}:=\left\{\begin{array}{cc}{t}^{min\{a,b\}},\hfill & t<1\hfill \\ t^{max\{a,b\}},\hfill & t\ge 1\hfill \end{array}\right.;t^{a\wedge b}:=\left\{\begin{array}{cc}{t}^{max\{a,b\}},\hfill & t<1\hfill \\ t^{min\{a,b\}},\hfill & t\ge 1\hfill \end{array}\right.$$

For any $h\in C_{+}\left(\overline{\Omega }\right)$, we define the variable-exponent Lebesgue space by

$$L^{h\left(x\right)}(\Omega )=\left\{u\mid u:\Omega \to \mathbb{R}\mathrm{is}\mathrm{measurable},{\int }_{\Omega }{\left|u\left(x\right)\right|}^{h\left(x\right)}dx<\infty \right\}.$$

Then, $L^{h\left(x\right)}(\Omega )$, endowed with the norm

$${\left|u\right|}_{h\left(x\right)}=inf\left\{\lambda >0:{\int }_{\Omega }{\left|{\displaystyle \frac{u\left(x\right)}{\lambda }}\right|}^{h\left(x\right)}dx\le 1\right\},$$

becomes a Banach space.

The convex functional $\rho :L^{h\left(x\right)}(\Omega )\to \mathbb{R}$ defined by

$$\rho \left(u\right)={\int }_{\Omega }{\left|u\left(x\right)\right|}^{h\left(x\right)}dx$$

is called modular on $L^{h\left(x\right)}(\Omega )$.

Proposition 1. 

If $u,u_n\in L^{h\left(x\right)}(\Omega )$, we have

(i) 

${\left|u\right|}_{h\left(x\right)}<1(=1;>1)\iff \rho \left(u\right)<1(=1;>1);$

(ii) 

${\left|u\right|}_{h\left(x\right)}>1\Rightarrow {\left|u\right|}_{h\left(x\right)}^{h^{-}}\le \rho \left(u\right)\le {\left|u\right|}_{h\left(x\right)}^{h^{+}}$;

${\left|u\right|}_{h\left(x\right)}\le 1\Rightarrow {\left|u\right|}_{h\left(x\right)}^{h^{+}}\le \rho \left(u\right)\le {\left|u\right|}_{h\left(x\right)}^{h^{-}};$

(iii) 

$\underset{n\to \infty }{lim}{|u_n-u|}_{h\left(x\right)}=0\iff \underset{n\to \infty }{lim}\rho (u_n-u)=0$.

Proposition 2. 

Let $h_1\left(x\right)$ and $h_2\left(x\right)$ be measurable functions such that $h_1\in L^{\infty }(\Omega )$ and $1\le h_1\left(x\right)h_2\left(x\right)\le \infty$ for a.e. $x\in \Omega$. Let $u\in L^{h_2\left(x\right)}(\Omega ),u\ne 0$. Then,

(i) 

${\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}\le 1⟹{\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}^{h_1^{+}}\le {\left|{\left|u\right|}^{h_1\left(x\right)}\right|}_{h_2\left(x\right)}\le {\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}^{h^{-}}$

(ii) 

${\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}>1⟹{\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}^{h_1^{-}}\le {\left|{\left|u\right|}^{h_1\left(x\right)}\right|}_{h_2\left(x\right)}\le {\left|u\right|}_{h_1\left(x\right)h_2\left(x\right)}^{h_1^{+}}$

(iii) 

In particular, if $h_1\left(x\right)=h$ is constant, then

$${\left|{\left|u\right|}^h\right|}_{h_2\left(x\right)}={\left|u\right|}_{h{h}_2\left(x\right)}^h.$$

The variable-exponent Sobolev space $W^{1,h\left(x\right)}(\Omega )$ is defined by

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

with the norm

$${\parallel u\parallel }_{1,h\left(x\right)}={\left|u\right|}_{h\left(x\right)}+{|\nabla u|}_{h\left(x\right)},\forall u\in W^{1,h\left(x\right)}(\Omega ).$$

where $\nabla u=({\partial }_{x_1}u,...,{\partial }_{x_N}u)$ and ${\partial }_{x_i}={\displaystyle \frac{\partial }{\partial x_i}}$ is the partial differential operator.

Proposition 3. 

If $1<h^{-}\le h^{+}<\infty$, then the spaces $L^{h\left(x\right)}(\Omega )$ and $W^{1,h\left(x\right)}(\Omega )$ are separable and reflexive Banach spaces.

The space $W_0^{1,h\left(x\right)}(\Omega )$ is defined as ${\overline{C_0^{\infty }(\Omega )}}^{{\parallel ·\parallel }_{1,h\left(x\right)}}=W_0^{1,h\left(x\right)}(\Omega )$; hence, it is the smallest closed set that contains $C_0^{\infty }(\Omega )$. Therefore, $W_0^{1,h\left(x\right)}(\Omega )$ is also a separable and reflexive Banach space due to the inclusion of $W_0^{1,h\left(x\right)}(\Omega )\subset W^{1,h\left(x\right)}(\Omega )$.

Note that, as a consequence of the Poincaré inequality, ${\parallel u\parallel }_{1,h\left(x\right)}$ and ${|\nabla u|}_{h\left(x\right)}$ are equivalent norms on $W_0^{1,h\left(x\right)}(\Omega )$. Therefore, for any $u\in W_0^{1,h\left(x\right)}(\Omega )$, we can define an equivalent norm $\parallel u\parallel$ such that

$$\parallel u\parallel ={|\nabla u|}_{h\left(x\right)}.$$

Proposition 4. 

Let $m\in C\left(\overline{\Omega }\right)$. If $1\le m\left(x\right)<h^{*}\left(x\right)$ for all $x\in \overline{\Omega }$, then the embeddings $W^{1,h\left(x\right)}(\Omega )\hookrightarrow L^{h\left(x\right)}(\Omega )$ and $W_0^{1,h\left(x\right)}(\Omega )\hookrightarrow L^{h\left(x\right)}(\Omega )$ are compact and continuous, where $h^{*}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{Nh\left(x\right)}{N-h\left(x\right)}}& \mathit{if}h\left(x\right)<N,\\ +\infty & \mathit{if}h\left(x\right)\ge N.\end{array}\right.$

Throughout the paper, we assume the following:

$\left(H_1\right)$

$p,q\in C_{+}\left(\overline{\Omega }\right)$ with $p^{-}\le p\left(x\right)\le p^{+}<q^{-}\le q\left(x\right)\le q^{+}<N$.

$\left(H_2\right)$

$\mu \in L^{\infty }(\Omega )$ such that $\mu (·)\ge 0$.

To address problem ($𝒫$), it is necessary to utilize the theory of the Musielak–Orlicz Sobolev space $W_0^{1,\mathcal{H}}(\Omega )$. Therefore, we subsequently introduce the double-phase operator, the Musielak–Orlicz space, and the Musielak–Orlicz Sobolev space in turn.

Let $\mathcal{H}:\Omega \times [0,\infty ]\to [0,\infty ]$ be the nonlinear function, i.e., the double-phase operator, defined by

$$\mathcal{H}(x,t)=t^{p\left(x\right)}+\mu \left(x\right)t^{q\left(x\right)}\mathrm{for}\mathrm{all}(x,t)\in \Omega \times [0,\infty ].$$

Then, the corresponding modular ${\rho }_{\mathcal{H}}(·)$ is given by

$${\displaystyle {\rho }_{\mathcal{H}}\left(u\right)={\int }_{\Omega }\mathcal{H}(x,|u\left|\right)dx={\int }_{\Omega }\left({\left|u\right|}^{p\left(x\right)}+\mu \left(x\right){\left|u\right|}^{q\left(x\right)}\right)dx.}$$

The Musielak–Orlicz space $L^{\mathcal{H}}(\Omega )$ is defined by

$$L^{\mathcal{H}}(\Omega )=\left\{u:\Omega \to \mathbb{R}\mathrm{measurable};{\rho }_{\mathcal{H}}\left(u\right)<+\infty \right\},$$

endowed with the Luxemburg norm

$${\parallel u\parallel }_{\mathcal{H}}=inf\left\{\zeta >0:{\rho }_{\mathcal{H}}\left({\displaystyle \frac{u}{\zeta }}\right)\le 1\right\}.$$

Analogously to Proposition 1, there are similar relationships between the modular ${\rho }_{\mathcal{H}}(·)$ and the norm ${\parallel ·\parallel }_{\mathcal{H}}$; see [23] (Proposition 2.13) for a detailed proof.

Proposition 5. 

Assume that $\left(H_1\right)$ holds, and $u\in L^{\mathcal{H}}(\Omega )$. Then,

$\left(i\right)$

If $u\ne 0$, then ${\parallel u\parallel }_{\mathcal{H}}=\zeta \iff {\rho }_{\mathcal{H}}\left({\displaystyle \frac{u}{\zeta }}\right)=1$;

$\left(ii\right)$

${\parallel u\parallel }_{\mathcal{H}}<1(\mathit{resp}.>1,=1)\iff {\rho }_{\mathcal{H}}\left({\displaystyle \frac{u}{\zeta }}\right)<1(\mathit{resp}.>1,=1)$;

$\left(iii\right)$

If ${\parallel u\parallel }_{\mathcal{H}}<1\Rightarrow {\parallel u\parallel }_{\mathcal{H}}^{q^{+}}\le {\rho }_{\mathcal{H}}\left(u\right)\le {\parallel u\parallel }_{\mathcal{H}}^{p^{-}}$;

$\left(iv\right)$

If ${\parallel u\parallel }_{\mathcal{H}}>1\Rightarrow {\parallel u\parallel }_{\mathcal{H}}^{p^{-}}\le {\rho }_{\mathcal{H}}\left(u\right)\le {\parallel u\parallel }_{\mathcal{H}}^{q^{+}}$;

$\left(v\right)$

${\parallel u\parallel }_{\mathcal{H}}\to 0\iff {\rho }_{\mathcal{H}}\left(u\right)\to 0$;

$\left(vi\right)$

${\parallel u\parallel }_{\mathcal{H}}\to +\infty \iff {\rho }_{\mathcal{H}}\left(u\right)\to +\infty$;

$\left(vii\right)$

${\parallel u\parallel }_{\mathcal{H}}\to 1\iff {\rho }_{\mathcal{H}}\left(u\right)\to 1$;

$\left(viii\right)$

If $u_n\to u$ in $L^{\mathcal{H}}(\Omega )$, then ${\rho }_{\mathcal{H}}\left(u_n\right)\to {\rho }_{\mathcal{H}}\left(u\right)$.

The Musielak–Orlicz Sobolev space $W^{1,\mathcal{H}}(\Omega )$ is defined by

$$W^{1,\mathcal{H}}(\Omega )=\left\{u\in L^{\mathcal{H}}(\Omega ):|\nabla u|\in L^{\mathcal{H}}(\Omega )\right\}$$

and equipped with the norm

$${\parallel u\parallel }_{1,\mathcal{H}}={\parallel \nabla u\parallel }_{\mathcal{H}}+{\parallel u\parallel }_{\mathcal{H}},$$

where ${\parallel \nabla u\parallel }_{\mathcal{H}}={\parallel |\nabla u|\parallel }_{\mathcal{H}}$. The space $W_0^{1,\mathcal{H}}(\Omega )$ is defined as ${\overline{C_0^{\infty }(\Omega )}}^{{\parallel ·\parallel }_{1,\mathcal{H}}}=W_0^{1,\mathcal{H}}(\Omega )$. Notice that $L^{\mathcal{H}}(\Omega ),W^{1,\mathcal{H}}(\Omega )$ and $W_0^{1,\mathcal{H}}(\Omega )$ are uniformly convex and reflexive Banach spaces. Moreover, we have the following embeddings [23] (Proposition 2.16).

Proposition 6. 

Let $\left(H\right)$ be satisfied. Then, the following embeddings hold:

$\left(i\right)$

$L^{\mathcal{H}}(\Omega )\hookrightarrow L^{h(·)}(\Omega ),W^{1,\mathcal{H}}(\Omega )\hookrightarrow W^{1,h(·)}(\Omega )$, $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow W_0^{1,h(·)}(\Omega )$ are continuous for all $h\in C\left(\overline{\Omega }\right)$ with $1\le h\left(x\right)\le p\left(x\right)$ for all $x\in \overline{\Omega }$.

$\left(ii\right)$

$W^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{h(·)}(\Omega )$ and $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{h(·)}(\Omega )$ are compact for all $h\in C\left(\overline{\Omega }\right)$ with $1\le h\left(x\right)<p^{*}\left(x\right)$ for all $x\in \overline{\Omega }$.

As the conclusion of Proposition 6, we have the continuous embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{h(·)}(\Omega )$, and $c_{\mathcal{H}}$ denotes the best constant such that

$${\left|u\right|}_{h(·)}\le c_{\mathcal{H}}{\parallel u\parallel }_{1,\mathcal{H},0}.$$

Moreover, by [23] (Proposition 2.18), $W_0^{1,\mathcal{H}}(\Omega )$ is compactly embedded in $L^{\mathcal{H}}(\Omega )$. Thus, $W_0^{1,\mathcal{H}}(\Omega )$ can be equipped with the equivalent norm

$${\parallel u\parallel }_{1,\mathcal{H},0}={\parallel \nabla u\parallel }_{\mathcal{H}}.$$

Proposition 7. 

For the convex functional ${\varrho }_{\mathcal{H}}\left(u\right):={\int }_{\Omega }\left({\displaystyle \frac{{|\nabla u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{{|\nabla u|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx$, we have ${\varrho }_{\mathcal{H}}\in C^1(W_0^{1,\mathcal{H}}(\Omega ),\mathbb{R})$ with the derivative

$$\langle {\varrho }_{\mathcal{H}}^{\prime }\left(u\right),\phi \rangle ={\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u)·\nabla \phi dx,$$

for all $u,\phi \in W_0^{1,\mathcal{H}}(\Omega )$, where $\langle ·,·\rangle$ is the dual pairing between $W_0^{1,\mathcal{H}}(\Omega )$ and its dual $W_0^{1,\mathcal{H}}{(\Omega )}^{*}$ [23].

Remark 1. 

Notice that

$\left(i\right)$

If ${\parallel \nabla u\parallel }_{\mathcal{H}}<1\Rightarrow {\displaystyle \frac{1}{q^{+}}}{\parallel \nabla u\parallel }_{\mathcal{H}}^{q^{+}}\le {\varrho }_{\mathcal{H}}\left(u\right)\le {\displaystyle \frac{1}{p^{-}}}{\parallel \nabla u\parallel }_{\mathcal{H}}^{p^{-}}$;

$\left(ii\right)$

If ${\parallel \nabla u\parallel }_{\mathcal{H}}>1\Rightarrow {\displaystyle \frac{1}{q^{+}}}{\parallel \nabla u\parallel }_{\mathcal{H}}^{p^{-}}\le {\varrho }_{\mathcal{H}}\left(u\right)\le {\displaystyle \frac{1}{p^{-}}}{\parallel \nabla u\parallel }_{\mathcal{H}}^{q^{+}}$.

We lastly introduce the seminormed space

$$L_{\mu }^{q(·)}(\Omega )=\left\{u:\Omega \to \mathbb{R}\mathrm{measurable};{\int }_{\Omega }\mu \left(x\right){\left|u\right|}^{q\left(x\right)}dx<+\infty \right\},$$

(3)

which is endowed with the seminorm

$${\left|u\right|}_{q(·),\mu }=inf\left\{\varsigma >0:{\int }_{\Omega }\mu \left(x\right){\left({\displaystyle \frac{\left|u\right|}{\varsigma }}\right)}^{q\left(x\right)}dx\le 1\right\}.$$

(4)

We have $L^{\mathcal{H}}(\Omega )\hookrightarrow L_{\mu }^{q(·)}(\Omega )$ continuously [23] (Proposition 2.16).

Proposition 8 

([24,25]). Let X be a vector space, and let $I:X\to \mathbb{R}$. Then, I is convex if and only if

$$I\left(\right(1-\lambda )u+\lambda v)\le (1-\lambda )\theta +\lambda \delta ,0<\lambda <1,$$

(5)

whenever $I\left(u\right)<\theta$ and $I\left(v\right)<\delta$, for all $u,v\in X$ and $\theta ,\delta \in \mathbb{R}$.

Proof. 

The proof was originally established by the author in his earlier work (see [24,25]). Assume that functional $I:X\to \mathbb{R}$ is convex. Since I is a real-valued functional, there are real numbers $\theta ,\delta \in \mathbb{R}$ such that $I\left(u\right)<\theta$ and $I\left(v\right)<\delta$. Then,

$$I\left(\right(1-\lambda )u+\lambda v)<(1-\lambda )I\left(u\right)+\lambda I\left(v\right)<(1-\lambda )\theta +\lambda \delta ,0<\lambda <1.$$

On the other hand, assume that (5) holds. Since $I\left(u\right)<\theta$ and $I\left(v\right)<\delta$, we can write

$$I\left(u\right)<I\left(u\right)+\epsilon :=\theta ,$$

$$I\left(v\right)<I\left(v\right)+\epsilon :=\delta ,$$

for a real number $\epsilon >0$. Therefore,

$$I\left(\right(1-\lambda )u+\lambda v)<(1-\lambda )I\left(u\right)+\lambda I\left(v\right)+\epsilon ,0<\lambda <1.$$

(6)

However, considering that $\epsilon >0$ is arbitrary, we conclude that

$$I\left(\right(1-\lambda )u+\lambda v)\le (1-\lambda )I\left(u\right)+\lambda I\left(v\right).$$

□

Proposition 9. 

Let $x,y\in {\mathbb{R}}^N$ and let $|·|$ be the Euclidean norm in ${\mathbb{R}}^N$. Then, for any $1\le p<\infty$ and the real parameters $a,b>0$, it holds that

$${\displaystyle \frac{{\left|a\right|x|}^{p-2}x-{b\left|y\right|}^{p-2}y|}{{\left|\right|x|}^{p-2}x-{\left|y\right|}^{p-2}y|}}\le \left(a+|a-b|\right)+{\displaystyle \frac{|a-b|}{{\left|\right|x|}^{p-2}x-{\left|y\right|}^{p-2}y|}}.$$

(7)

Proof. 

If $a=b$, then there is nothing to do. Thus, we assume that $a\ne b$.

Put

$$\Lambda (x,y)={\displaystyle \frac{{\left|a\right|x|}^{p-2}x-{b\left|y\right|}^{p-2}y|}{{\left|\right|x|}^{p-2}x-{\left|y\right|}^{p-2}y|}}.$$

(8)

Notice that $\Lambda$ is invariant by any orthogonal transformation T—that is, $\Lambda (Tx,Ty)=\Lambda (x,y)$ for all $x,y\in {\mathbb{R}}^N$.Thus, using this fact, along with the homogeneity of $\Lambda$, we can let $x=\left|x\right|e_1$ and assume that $x=e_1$. Then, without loss of generality, it is enough to work with the function

$$\Lambda (e_1,y)={\displaystyle \frac{|a{e}_1-{b\left|y\right|}^{p-2}y|}{|e_1-{\left|y\right|}^{p-2}y|}}.$$

(9)

In doing so, first, we have

$$\begin{array}{cc}\hfill |a{e}_1-{b\left|y\right|}^{p-2}y|& =a|e_1-{\displaystyle \frac{b}{a}}{\left|y\right|}^{p-2}y|=a|(e_1-{\left|y\right|}^{p-2}y)+\left(1-{\displaystyle \frac{b}{a}}\right){\left|y\right|}^{p-2}y|\hfill \\ \hfill & \le a|e_1-{\left|y\right|}^{p-2}{y|+|a-b\left|\right|\left|y\right|}^{p-2}y|\hfill \\ \hfill & \le a|e_1-{\left|y\right|}^{p-2}y|+|a-b|\left(|e_1-{\left|y\right|}^{p-2}y|+|e_1|\right)\hfill \\ \hfill & \le a|e_1-{\left|y\right|}^{p-2}y|+|a-b\left|\right|e_1-{\left|y\right|}^{p-2}y|+|a-b|\hfill \\ \hfill & \le |e_1-{\left|y\right|}^{p-2}y|\left(a+|a-b|\right)+|a-b|.\hfill \end{array}$$

(10)

Then, using this in (9), we obtain

$$\begin{array}{cc}\hfill \Lambda (e_1,y)& \le {\displaystyle \frac{|e_1-{\left|y\right|}^{p-2}y|\left(a+|a-b|\right)+|a-b|}{|e_1-{\left|y\right|}^{p-2}y|}}\hfill \\ \hfill & \le a+|a-b|+{\displaystyle \frac{|a-b|}{|e_1-{\left|y\right|}^{p-2}y|}}\hfill \end{array}$$

(11)

which shows that (7) holds. □

3. The Main Results

The energy functional $\mathcal{I}:W_0^{1,\mathcal{H}}(\Omega )\to \mathbb{R}$ corresponding to equation ($𝒫$) by

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\lambda }\left(u\right)& =\widehat{\mathcal{M}}\left({\int }_{\Omega }\left({\displaystyle \frac{{|\nabla u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{{|\nabla u|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx\right)-\lambda {\int }_{\Omega }F(x,u)dx,\hfill \end{array}$$

(12)

where $F(x,t)={\int }_0^{t}f(x,s)ds$, and $\widehat{\mathcal{M}}\left(t\right)={\int }_0^t\mathcal{M}\left(s\right)ds$.

Definition 1. 

A function $u\in W_0^{1,\mathcal{H}}(\Omega )$ is called a weak solution to problem ($𝒫$) if

$$\begin{array}{cc}\hfill & {\mathcal{M}}_{\varrho }\left(u\right){\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u)·\nabla \phi dx\hfill \\ \hfill & =\lambda {\int }_{\Omega }f(x,u)\phi dx,\forall \phi \in W_0^{1,\mathcal{H}}(\Omega ),\hfill \end{array}$$

(13)

where $\mathcal{M}\left({\varrho }_{\mathcal{H}}\left(u\right)\right):={\mathcal{M}}_{\varrho }\left(u\right)$. It is well known that the critical points of the functional ${\mathcal{I}}_{\lambda }$ correspond to the weak solutions of problem ($𝒫$).

Let us define the functionals $\mathcal{J},\mathcal{K}:W_0^{1,\mathcal{H}}(\Omega )\to \mathbb{R}$ by

$$\mathcal{J}\left(u\right):={\int }_{\Omega }F(x,u)dx,$$

(14)

and

$$\mathcal{K}\left(u\right):=\widehat{\mathcal{M}}\left({\varrho }_{\mathcal{H}}\left(u\right)\right)={\widehat{\mathcal{M}}}_{\varrho }\left(u\right).$$

(15)

Then,

$${\mathcal{I}}_{\lambda }(·):=\mathcal{K}(·)-\lambda \mathcal{J}(·).$$

(16)

To obtain the main result, we apply the following well-known critical point result given by Bonanno and and Marano.

Lemma 1 

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

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

(17)

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

(a1) 

${\displaystyle \frac{1}{r}}{sup}_{\Phi \left(x\right)\le r}\Psi \left(x\right)\le {\displaystyle \frac{\Psi \left(\overline{x}\right)}{\Phi \left(\overline{x}\right)}}$;

(a2) 

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

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

We assume the following:

$\left(f_1\right)$

$f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function and there exist real parameters ${\overline{c}}_1,{\overline{c}}_2\ge 0$ satisfying

$$\left|f(x,t)\right|\le {\overline{c}}_1+{\overline{c}}_2{\left|t\right|}^{s\left(x\right)-1},\forall (x,t)\in \Omega \times \mathbb{R}$$

(18)

where $s\in C\left(\overline{\Omega }\right)$ with $1<s\left(x\right)<p^{*}\left(x\right)$ for all $x\in \overline{\Omega }$;

$\left(f_2\right)$

There exists a real parameter ${\overline{c}}_3\ge 0$ and $r\in C_{+}\left(\overline{\Omega }\right)$ with $r^{-}\le r\left(x\right)\le r^{+}<p^{-}$ satisfying

$$F(x,t)\le {\overline{c}}_3{(1+|t|}^{r\left(x\right)}),\forall (x,t)\in \Omega \times \mathbb{R};$$

(19)

$\left(f_3\right)$

$F(x,t)\ge 0$ for all $(x,t)\in \Omega \times {\mathbb{R}}_{+}$;

$\left(M\right)$

$\mathcal{M}:(0,\infty )\to [m_0,m^0)$ is a $C^1$-continuous nondecreasing function satisfying

$${\kappa }_1{t}^{{\alpha }_1-1}\le \mathcal{M}\left(t\right)\le {\kappa }_2{t}^{{\alpha }_2-1},$$

(20)

where $m_0,m^0,{\kappa }_1,{\kappa }_2,{\alpha }_1,{\alpha }_2$ are positive real parameters such that ${\kappa }_2\ge {\kappa }_1$ and ${\alpha }_2\ge {\alpha }_1>1$.

The main result of the paper is given in the following.

Theorem 1. 

Assume that $\left(f_1\right)-\left(f_3\right)$ and $\left(M\right)$ are satisfied. Assume also that

$\left(f_4\right)$

There exist positive real parameters $r,\delta$ with

$r<{\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\omega }_N^{{\alpha }_1}{R}^{N{\alpha }_1}{(2^N-1)}^{{\alpha }_1}{2}^{-N{\alpha }_1}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{{\alpha }_1(p^{-}\wedge p^{+})}$ satisfying

$$\begin{array}{cc}\hfill {\sigma }_r:& =\{{\overline{c}}_1{c}_{\mathcal{H}}{\left(q^{+}\right)}^{{\displaystyle \frac{1}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}+{\overline{c}}_2{c}_{\mathcal{H}}^{s^{+}}{\left(q^{+}\right)}^{{\displaystyle \frac{s^{+}}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}}\}\hfill \\ \hfill & <{\displaystyle \frac{2^{N({\alpha }_2-1)}{\alpha }_2{\left(p^{-}\right)}^{{\alpha }_2}{inf}_{x\in \Omega }F(x,\delta )}{{\kappa }_2{(1+|\mu |}_{\infty }{)}^{{\alpha }_2}{\omega }_N^{{\alpha }_2-1}{(2^N-1)}^{{\alpha }_2}{R}^{N({\alpha }_2-1)}{\left(\frac{2\delta }{R}\right)}^{{\alpha }_2(p^{-}\vee q^{+})}}}:={\sigma }^r.\hfill \end{array}$$

(21)

Then, for any $\lambda \in {\lambda }_{r,\delta }:=\left({\displaystyle \frac{1}{{\sigma }^r}},{\displaystyle \frac{1}{{\sigma }_r}}\right)$, the problem ($𝒫$) admits at least three distinct weak solutions.

First, we obtain the regularity results of the functionals ${\mathcal{I}}_{\lambda },\mathcal{K}$, and $\mathcal{J}$, which are needed to apply Lemma 1.

Lemma 2. 

Assume that $\left(M\right)$ holds. Then, the following hold.

$\left(i\right)$

$\mathcal{K}$ is coercive.

$\left(ii\right)$

$\mathcal{K}$ is Gâteaux-differentiable and the Gâteaux derivative ${\mathcal{K}}^{\prime }:W_0^{1,\mathcal{H}}(\Omega )\to W_0^{1,\mathcal{H}}{(\Omega )}^{*}$ given by the formula for $u,\phi \in W_0^{1,p\left(x\right)}(\Omega )$

$$\langle {\mathcal{K}}^{\prime }\left(u\right),\phi \rangle ={\mathcal{M}}_{\varrho }\left(u\right){\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u)·\nabla \phi dx$$

(22)

is bounded and continuous.

$\left(iii\right)$

${\mathcal{K}}^{\prime }$ is strictly monotone.

$\left(iv\right)$

${\mathcal{K}}^{\prime }$ satisfies the $\left(S_{+}\right)$-property—that is,

$$\mathit{if}{u}_n\rightharpoonup u\in W_0^{1,\mathcal{H}}(\Omega )\mathit{and}\underset{n\to \infty }{lim\; sup}\langle {\mathcal{K}}^{\prime }\left(u_n\right),u_n-u\rangle \le 0,\mathit{then}{u}_n\to u\in W_0^{1,\mathcal{H}}(\Omega ).$$

$\left(v\right)$

${\mathcal{K}}^{\prime }$ is coercive and a homeomorphism.

$\left(vi\right)$

$\mathcal{K}$ is sequentially weakly lower semicontinuous.

Proof. 

(i) Using $\left(M\right)$ and Proposition 5, it reads

$$\mathcal{K}\left(u\right)={\int }_0^{{\varrho }_{\mathcal{H}}\left(u\right)}\mathcal{M}\left(s\right)ds\ge {\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{{\alpha }_1{p}^{-}},$$

(23)

which implies that $\mathcal{K}$ is coercive.

(ii) Using the mean value theorem, it reads

$$\begin{array}{cc}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right),\phi \rangle & \\ \hfill =& \underset{t\to 0}{lim}({\mathcal{M}}_{\varrho }(u+\delta t\phi )\times {\int }_{\Omega }({|\nabla (u+\delta t\phi )|}^{p\left(x\right)-2}\nabla (u+\delta t\phi )\hfill \\ \hfill & +{\mu \left(x\right)|\nabla (u+\delta t\phi )|}^{q\left(x\right)-2}\nabla (u+\delta t\phi ))·\nabla \phi dx),\hfill \end{array}$$

(24)

for all $u,\phi \in W_0^{1,\mathcal{H}}(\Omega )$, and $0\le \delta \le 1$.

Let

$$\begin{array}{cc}\hfill \Theta (u+\delta t\phi ):=& {\left(\right|\nabla (u+\delta t\phi )|}^{p\left(x\right)-2}\nabla (u+\delta t\phi )\hfill \\ \hfill & +{\mu \left(x\right)|\nabla (u+\delta t\phi ))|}^{q\left(x\right)-2}\nabla (u+\delta t\phi ))·\nabla \phi .\hfill \end{array}$$

(25)

Applying Young’s inequality, we obtain

$$\begin{array}{cc}\hfill |\Theta (u+\delta t\phi \left)\right|& \le \widehat{c}({|\nabla u|}^{p\left(x\right)}+{|\nabla \phi |}^{p\left(x\right)}+{\left|\mu \right|}_{\infty }^{q_{+}}{|\nabla u|}^{q\left(x\right)}+{|\nabla \phi |}^{q\left(x\right)}),\hfill \end{array}$$

(26)

where $\widehat{c}:={\displaystyle \frac{2^{q^{+}}(q^{+}-1)+1}{2q^{-}}}$. Thanks to the relation $L^{q\left(x\right)}(\Omega )\subset L^{p\left(x\right)}(\Omega )\subset L^1(\Omega )$, the right-hand side of (26) is summable over $\Omega$. Therefore, using assumption $\left(M\right)$ and the Lebesgue dominated convergence theorem together provides

$$\begin{array}{cc}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right),\phi \rangle & =\left(\underset{t\to 0}{lim}{\mathcal{M}}_{\varrho }(u+\delta t\phi )\times {\int }_{\Omega }\underset{t\to 0}{lim}{\left\{\right|\nabla (u+\delta t\phi )|}^{p\left(x\right)-2}\nabla (u+\delta t\phi )\right.\hfill \\ \hfill & \left.+{\mu \left(x\right)|\nabla (u+\delta t\phi ))|}^{q\left(x\right)-2}\nabla (u+\delta t\phi )\}·\nabla \phi dx\right)\hfill \\ \hfill & ={\mathcal{M}}_{\varrho }\left(u\right){\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u)·\nabla \phi dx.\hfill \end{array}$$

(27)

The linearity of Formula (27) follows due to the linearity of integration and the fact that it is linear in $\nabla \phi$.

Next, we show that ${\mathcal{K}}^{\prime }$ with Formula (27) is bounded. Then, using Hölder’s inequality, Proposition 2, Remark 1, and the involved embeddings, it follows that

$$\begin{array}{cc}\hfill |\langle {\mathcal{K}}^{\prime }(u),\phi \rangle |& =|{\mathcal{M}}_{\varrho }(u){\int }_{\Omega }{(|\nabla u|}^{p(x)-2}\nabla u+{\mu (x)|\nabla u|}^{q(x)-2}\nabla u)·\nabla \phi dx|\hfill \\ \hfill & \le {\displaystyle \frac{{\kappa }_2}{{(p^{-})}^{{\alpha }_2-1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{({\alpha }_2-1)p^{-}\vee q^{+}}\left({||\nabla u|}^{p(x)-1}{|}_{{\displaystyle \frac{p(x)}{p(x)-1}}}{|\nabla \phi |}_{p(x)}\right.\hfill \\ \hfill & \left.+{|\mu |}_{\infty }{||\nabla u|}^{q(x)-1}{|}_{{\displaystyle \frac{q(x)}{q(x)-1}}}{|\nabla \phi |}_{q(x)}\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\kappa }_2}{{(p^{-})}^{{\alpha }_2-1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{({\alpha }_2-1)p^{-}\vee q^{+}}\left({\parallel u\parallel }_{1,\mathcal{H},0}^{p_{+}-1}+{|\mu |}_{\infty }{\parallel u\parallel }_{1,\mathcal{H},0}^{q_{+}-1}\right){\parallel \phi \parallel }_{1,\mathcal{H},0}\hfill \\ \hfill & \le {\displaystyle \frac{{2|\mu |}_{\infty }{\kappa }_2}{{(p^{-})}^{{\alpha }_2-1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{\tau }{\parallel \phi \parallel }_{1,\mathcal{H},0}.\hfill \end{array}$$

(28)

where $\tau :=max\{({\alpha }_2-1)(p^{-}\vee q^{+})+(p_{+}-1),({\alpha }_2-1)(p^{-}\vee q^{+})+(q_{+}-1)\}$. Therefore,

$$\begin{array}{c}\hfill \parallel {\mathcal{K}}^{\prime }{\left(u\right)\parallel }_{W_0^{1,\mathcal{H}}{(\Omega )}^{*}}=\underset{\phi \in W_0^{1,\mathcal{H}}(\Omega ),{\parallel \phi \parallel }_{1,\mathcal{H},0}\le 1}{sup}|\langle {\mathcal{K}}^{\prime }\left(u\right),\phi \rangle |\le {\displaystyle \frac{{\left|\mu \right|}_{\infty }{\kappa }_2}{{\left(p^{-}\right)}^{{\alpha }_2-1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{\tau },\forall u\in W_0^{1,\mathcal{H}}(\Omega )\end{array}$$

(29)

and ${\mathcal{K}}^{\prime }$ is bounded. Therefore, $\mathcal{K}$ is Gâteaux-differentiable with the derivative given by Formula (27).

Next, we continue with the continuity of ${\mathcal{K}}^{\prime }$. In doing so, for a sequence $\left(u_n\right)\subset W_0^{1,\mathcal{H}}(\Omega )$, assume that $u_n\to u$ in $W_0^{1,\mathcal{H}}(\Omega )$. Then, using Proposition 9, we have

$$\begin{array}{cc}\hfill & |\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),v\rangle |\hfill \\ \hfill & \le {\int }_{\Omega }|{\mathcal{M}}_{\varrho }\left(u_n\right)|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{\mathcal{M}}_{\varrho }\left(u\right){|\nabla u|}^{p\left(x\right)-2}\nabla u||\nabla v|dx\hfill \\ \hfill & +{\int }_{\Omega }\mu \left(x\right)|{\mathcal{M}}_{\varrho }\left(u_n\right)|\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{\mathcal{M}}_{\varrho }\left(u\right){|\nabla u|}^{q\left(x\right)-2}\nabla u||\nabla v|dx\hfill \\ \hfill & \le {\int }_{\Omega }\left\{||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u|\times (G_n+{\mathcal{M}}_{\varrho }\left(u_n\right))+G_n\right\}|\nabla v|dx\hfill \\ \hfill & +{\int }_{\Omega }\mu \left(x\right)\left\{||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}\nabla u|\times (G_n+{\mathcal{M}}_{\varrho }\left(u_n\right))+G_n\right\}|\nabla v|dx\hfill \\ \hfill & \le (G_n+{\mathcal{M}}_{\varrho }\left(u_n\right)){\int }_{\Omega }||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u||\nabla v|dx+G_n{\int }_{\Omega }|\nabla v|dx\hfill \\ \hfill & +(G_n+{\mathcal{M}}_{\varrho }\left(u_n\right)){\int }_{\Omega }\mu \left(x\right)||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}\nabla u||\nabla v|dx+G_n{\int }_{\Omega }|\nabla v|dx.\hfill \end{array}$$

(30)

where $G_n:=|{\mathcal{M}}_{\varrho }\left(u_n\right)-{\mathcal{M}}_{\varrho }\left(u\right)|$. Now, if we apply Hölder’s inequality and consider the embedding $L^{\mathcal{H}}(\Omega )\hookrightarrow L_{\mu }^{q\left(x\right)}(\Omega )$ and Propositions 5 and 6, it reads

$$\begin{array}{cc}\hfill & |\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),v\rangle |\hfill \\ \hfill & \le (G_n+{\mathcal{M}}_{\varrho }\left(u_n\right))|||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u|{|}_{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}{|\nabla v|}_{p\left(x\right)}+G_n{|\Omega ||\nabla v|}_{p\left(x\right)}\hfill \\ \hfill & +(G_n+{\mathcal{M}}_{\varrho }\left(u_n\right))|\mu {\left(x\right)}^{{\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}}||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}\nabla u|{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}|\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}{|\nabla v||}_{q\left(x\right)}\hfill \\ \hfill & +G_n{|\Omega ||\nabla v|}_{p\left(x\right)},\hfill \end{array}$$

(31)

and, therefore,

$$\begin{array}{c}\hfill \parallel {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right){\parallel }_{W_0^{1,\mathcal{H}}{(\Omega )}^{*}}=\underset{v\in W_0^{1,\mathcal{H}}(\Omega ),{\parallel v\parallel }_{1,\mathcal{H},0}\le 1}{sup}\left|\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),v\rangle \right|\to 0.\end{array}$$

(32)

Note that the result (32) follows due to the following.

Since $u_n\to u$ in $W_0^{1,\mathcal{H}}(\Omega )$, by the embeddings $L^{\mathcal{H}}(\Omega )\hookrightarrow L_{\mu }^{q\left(x\right)}(\Omega )$, $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\mathcal{H}}(\Omega )$ and $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{p\left(x\right)}(\Omega )$, we have

$$\underset{n\to \infty }{lim}{\int }_{\Omega }|\nabla u_n{|}^{p\left(x\right)}dx={\int }_{\Omega }{|\nabla u|}^{p\left(x\right)}dx,$$

(33)

and

$$\underset{n\to \infty }{lim}{\int }_{\Omega }\mu \left(x\right)|\nabla u_n{|}^{q\left(x\right)}dx={\int }_{\Omega }\mu \left(x\right){|\nabla u|}^{q\left(x\right)}dx.$$

(34)

However, by Vitali’s Theorem [26] (Theorem 4.5.4), (33) and (34) mean $|\nabla u_n|\to |\nabla u|$ and $\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}|\nabla u_n{|\to \mu \left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}|\nabla u|$ in measure in $\Omega$ and the sequences ${\left\{|\nabla u_n{|}^{p\left(x\right)}\right\}}_n$ and ${\left\{\mu \left(x\right)|\nabla u_n{|}^{q\left(x\right)}\right\}}_n$ are uniformly integrable over $\Omega$. Now, let us consider the inequalities

$$||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}{\nabla u|}^{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\le 2^{{\displaystyle \frac{p^{+}}{p^{-}-1}}-1}(|\nabla u_n{|}^{p\left(x\right)}+{|\nabla u|}^{p\left(x\right)}),$$

(35)

and

$$\mu \left(x\right)||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}{\nabla u|}^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\le 2^{{\displaystyle \frac{q^{+}}{q^{-}-1}}-1}\mu \left(x\right)(|\nabla u_n{|}^{q\left(x\right)}+{|\nabla u|}^{q\left(x\right)}).$$

(36)

Therefore, the families ${\left\{||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}{\nabla u|}^{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\right\}}_n$ and ${\left\{\mu \left(x\right)||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}{\nabla u|}^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\right\}}_n$ are also uniformly integrable over $\Omega$. Thus, by Vitali’s Theorem, ${|\nabla u|}^{p\left(x\right)-2}\nabla u$ and ${\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u$ are integrable and $|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n\to {|\nabla u|}^{p\left(x\right)-2}\nabla u$ and $\mu \left(x\right)|\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n\to \mu \left(x\right){|\nabla u|}^{q\left(x\right)-2}\nabla u$ in $L^1(\Omega )$. Hence,

$$|||\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{|\nabla u|}^{p\left(x\right)-2}\nabla u|{|}_{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\to 0,$$

(37)

and

$$|\mu {\left(x\right)}^{{\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}}||\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{|\nabla u|}^{q\left(x\right)-2}\nabla u|{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\to 0.$$

(38)

Lastly, by assumption $\left(M\right)$ and Proposition 7, $(G_n+{\mathcal{M}}_{\varrho }\left(u_n\right))$ is bounded for $n=1,2,...$, and

$$G_n=|{\mathcal{M}}_{\varrho }\left(u_n\right)-{\mathcal{M}}_{\varrho }\left(u\right)|\to 0\mathrm{as}n\to \infty .$$

(39)

Therefore, the result (32) follows. Putting all these together, we infer that ${\mathcal{K}}^{\prime }$ is continuously Gâteaux-differentiable and the derivative is given by Formula (22).

(iii) Now, we show that ${\mathcal{K}}^{\prime }$ is strictly monotone. To do so, we argue similarly to [27]. Let $u,v\in W_0^{1,\mathcal{H}}(\Omega )$ with $u\ne v$. Without loss of generality, we can assume that ${\varrho }_{\mathcal{H}}\left(u\right)\ge {\varrho }_{\mathcal{H}}\left(v\right)$. Then, ${\mathcal{M}}_{\varrho }\left(u\right)\ge {\mathcal{M}}_{\varrho }\left(v\right)$ due to $\left(M\right)$ and Proposition 7.

Noticing that

$$0\le {(\nabla u-\nabla v)}^2\Rightarrow \nabla u·\nabla v\le 2^{-1}{\left(\right|\nabla u|}^2+{|\nabla v|}^2),$$

(40)

we obtain

$$\begin{array}{cc}\hfill \langle {\varrho }_{\mathcal{H}}^{\prime }\left(u\right),u-v\rangle & ={\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}\nabla u)·\nabla (u-v)dx\hfill \\ \hfill & ={\int }_{\Omega }\left({|\nabla u|}^{p\left(x\right)}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)}-{\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2})\nabla u·\nabla v\right)dx\hfill \\ \hfill & \ge 2^{-1}{\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}{\left)\right(|\nabla u|}^2-{|\nabla v|}^2)dx,\hfill \end{array}$$

(41)

and

$$\begin{array}{cc}\hfill \langle {\varrho }_{\mathcal{H}}^{\prime }\left(v\right),v-u\rangle & ={\int }_{\Omega }{\left(\right|\nabla v|}^{p\left(x\right)-2}\nabla v+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2}\nabla v)·\nabla (v-u)dx\hfill \\ \hfill & ={\int }_{\Omega }\left({|\nabla v|}^{p\left(x\right)}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)}-{\left(\right|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2})\nabla u·\nabla v\right)dx\hfill \\ \hfill & \ge 2^{-1}{\int }_{\Omega }{\left(\right|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2}{\left)\right(|\nabla v|}^2-{|\nabla u|}^2)dx.\hfill \end{array}$$

(42)

Next, we partition $\Omega$ into ${\Omega }_{\ge }=\{x\in \Omega :|\nabla u|\ge |\nabla v|\}$ and ${\Omega }_{<}=\{x\in \Omega :|\nabla u|<|\nabla v\left|\right\}$. Hence, using (41), (42), and $\left(M\right)$, we have

$$\begin{array}{cc}\hfill I_1\left(u\right)& :={\mathcal{M}}_{\varrho }\left(u\right)\langle {\varrho }_{\mathcal{H}}^{\prime }\left(u\right),u-v\rangle \hfill \\ \hfill & ={\mathcal{M}}_{\varrho }\left(u\right){\int }_{{\Omega }_{\ge }}\left({|\nabla u|}^{p\left(x\right)}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)}+{\mu \left(x\right)|\nabla u|}^{r\left(x\right)}-{\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2})\nabla u·\nabla v\right)dx\hfill \\ \hfill & \ge {\displaystyle \frac{{\mathcal{M}}_{\varrho }\left(u\right)}{2}}{\int }_{{\Omega }_{\ge }}{\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}{\left)\right(|\nabla u|}^2-{|\nabla v|}^2)dx\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{M}}_{\varrho }\left(v\right)}{2}}{\int }_{{\Omega }_{\ge }}{\left(\right|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2}{\left)\right(|\nabla u|}^2-{|\nabla v|}^2)dx\hfill \\ \hfill & \ge {\displaystyle \frac{{\mathcal{M}}_{\varrho }\left(v\right)}{2}}{\int }_{{\Omega }_{\ge }}\left({\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}{)-(|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2})\right)\hfill \\ \hfill & \times \left(\right|\nabla u{|}^2-|\nabla v{|}^2)dx\hfill \\ \hfill & \ge {\displaystyle \frac{m_0}{2}}{\int }_{{\Omega }_{\ge }}\left({\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)-2}{)-(|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2})\right)\hfill \\ \hfill & \times \left(\right|\nabla u{|}^2-|\nabla v{|}^2)dx\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

(43)

and, similarly,

$$\begin{array}{cc}\hfill I_2\left(v\right)& :={\mathcal{M}}_{\varrho }\left(v\right)\langle {\varrho }_{\mathcal{H}}^{\prime }\left(v\right),v-u\rangle \hfill \\ \hfill & ={\mathcal{M}}_{\varrho }\left(v\right){\int }_{{\Omega }_{<}}\left({|\nabla v|}^{p\left(x\right)}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)}-{\left(\right|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2})\nabla u·\nabla v\right)dx\hfill \\ \hfill & \ge {\displaystyle \frac{m_0}{2}}{\int }_{{\Omega }_{<}}\left({\left(\right|\nabla u|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2}{)-(|\nabla v|}^{p\left(x\right)-2}+{\mu \left(x\right)|\nabla v|}^{q\left(x\right)-2})\right)\hfill \\ \hfill & \times \left(\right|\nabla u{|}^2-|\nabla v{|}^2)dx\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

(44)

Note that

$$\begin{array}{c}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right)-{\mathcal{K}}^{\prime }\left(v\right),u-v\rangle =\langle {\mathcal{K}}^{\prime }\left(u\right),u-v\rangle +\langle {\mathcal{K}}^{\prime }\left(v\right),v-u\rangle =I_1\left(u\right)+I_2\left(v\right)\ge 0.\end{array}$$

(45)

However, we discard the case of $\langle {\mathcal{K}}^{\prime }\left(u\right)-{\mathcal{K}}^{\prime }\left(v\right),u-v\rangle =0$ since this would eventually imply that ${\partial }_{x_i}u={\partial }_{x_i}v$ for $i=1,2,...,N$, which would contradict the assumption that $u\ne v$ in $W_0^{1,\mathcal{H}}(\Omega )$. Thus,

$$\begin{array}{c}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right)-{\mathcal{K}}^{\prime }\left(v\right),u-v\rangle >0.\end{array}$$

(46)

(iv) Let $\left(u_n\right)\subset W_0^{1,\mathcal{H}}(\Omega )$ be a sequence satisfying

$$u_n\rightharpoonup u\in W_0^{1,\mathcal{H}}(\Omega ),$$

(47)

$$\underset{n\to \infty }{lim\; sup}\langle {\mathcal{K}}^{\prime }\left(u_n\right),u_n-u\rangle \le 0.$$

(48)

We shall show that $u_n\to u\in W_0^{1,\mathcal{H}}(\Omega )$.

By (47), we have

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

(49)

Then, by (48) and (49), it reads

$$\underset{n\to \infty }{lim\; sup}\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),u_n-u\rangle \le 0.$$

(50)

However, if we also consider that ${\mathcal{K}}^{\prime }$ is strictly monotone, we have

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

(51)

Therefore,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),u_n-u\rangle \hfill \\ \hfill & =\underset{n\to \infty }{lim}\left({\int }_{\Omega }\left({\mathcal{M}}_{\varrho }\left(u_n\right)|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n-{\mathcal{M}}_{\varrho }\left(u\right){|\nabla u|}^{p\left(x\right)-2}\nabla u\right)\nabla (u_n-u)dx\right.\hfill \\ \hfill & \left.+{\int }_{\Omega }\mu \left(x\right)\left({\mathcal{M}}_{\varrho }\left(u_n\right)|\nabla u_n{|}^{q\left(x\right)-2}\nabla u_n-{\mathcal{M}}_{\varrho }\left(u\right){|\nabla u|}^{q\left(x\right)-2}\nabla u\right)\nabla (u_n-u)dx\right)=0.\hfill \end{array}$$

(52)

Recall the inequality [28]

$${\left(\right|\xi |}^{h-2}\xi -{\left|\eta \right|}^{h-2}\eta )·(\xi -\eta )>0,\forall \xi ,\eta \in {\mathbb{R}}^N,h\ge 1.$$

(53)

Then, using $\left(M\right)$, (52), and (53) together implies that $\nabla u_n\to \nabla u$ and $u_n\to u$ in measure in $\Omega$. Then, by the Riesz theorem, there are subsequences, not relabeled, that converge pointwise a.e. on $\Omega$ to $\nabla u$ and u, respectively [29]. Hence, if we let

$${\gamma }_n\left(x\right):={\displaystyle \frac{|\nabla u_n{|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|\nabla u_n{|}^{q\left(x\right)}}{q\left(x\right)}},$$

(54)

and

$$\gamma \left(x\right):={\displaystyle \frac{{|\nabla u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{{|\nabla u|}^{q\left(x\right)}}{q\left(x\right)}},$$

(55)

then $|{\gamma }_n\left(x\right)-\gamma \left(x\right)|\to 0$ a.e. on $\Omega$.

On the other hand, applying Young’s inequality, it reads

$$\begin{array}{cc}\hfill \langle {\mathcal{K}}^{\prime }\left(u_n\right),u_n-u\rangle & ={\mathcal{M}}_{\varrho }\left(u_n\right){\int }_{\Omega }\left(|\nabla u_n{|}^{p\left(x\right)-2}\nabla u_n+\mu \left(x\right){|\nabla u_n|}^{q\left(x\right)-2}\nabla u_n\right)\nabla (u_n-u)dx\hfill \\ \hfill & \ge {\mathcal{M}}_{\varrho }\left(u_n\right){\int }_{\Omega }\left(|\nabla u_n{|}^{p\left(x\right)}+\mu \left(x\right){|\nabla u_n|}^{q\left(x\right)}\right)dx\hfill \\ \hfill & -{\mathcal{M}}_{\varrho }\left(u_n\right){\int }_{\Omega }\left(|\nabla u_n{|}^{p\left(x\right)-1}|\nabla u|+\mu \left(x\right)|\nabla u_n{|}^{q\left(x\right)-1}|\nabla u|\right)dx\hfill \\ \hfill & \ge {\mathcal{M}}_{\varrho }\left(u_n\right){\int }_{\Omega }\left({\displaystyle \frac{|\nabla u_n{|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|\nabla u_n{|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx\hfill \\ \hfill & -{\mathcal{M}}_{\varrho }\left(u_n\right){\int }_{\Omega }\left\{{\displaystyle \frac{p\left(x\right)-1}{p\left(x\right)}}|\nabla u_n{|}^{p\left(x\right)}+{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\right.\hfill \\ \hfill & \left.+\mu \left(x\right)\left({\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}|\nabla u_n{|}^{q\left(x\right)}+{\displaystyle \frac{1}{q\left(x\right)}}{|\nabla u|}^{q\left(x\right)}\right)\right\}dx\hfill \\ \hfill & ={\int }_{\Omega }{\mathcal{M}}_{\varrho }\left(u_n\right)({\gamma }_n\left(x\right)-\gamma \left(x\right))dx.\hfill \end{array}$$

(56)

Then, using (48), (56), Fataou’s lemma, and Proposition 7 together provides

$$\underset{n\to \infty }{lim}{\int }_{\Omega }{\mathcal{M}}_{\varrho }\left(u_n\right)({\gamma }_n\left(x\right)-\gamma \left(x\right))dx=0.$$

(57)

By assumption $\left(M\right)$, we infer from (57) that

$$\underset{n\to \infty }{lim}{\int }_{\Omega }{\gamma }_n\left(x\right)dx={\int }_{\Omega }\gamma \left(x\right)dx.$$

(58)

By the Vitali Convergence Theorem [26] (Theorem 4.5.4), (58) means $|\nabla u_n|\to |\nabla u|$ and $\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}|\nabla u_n{|\to \mu \left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}|\nabla u|$ in measure in $\Omega$ and the sequences ${\left\{|\nabla u_n{|}^{p\left(x\right)}\right\}}_n$ and ${\left\{\mu \left(x\right)|\nabla u_n{|}^{q\left(x\right)}\right\}}_n$ are uniformly integrable over $\Omega$. Hence, the function family ${\left\{{\gamma }_n\right\}}_n={\left\{{\displaystyle \frac{|\nabla u_n{|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|\nabla u_n{|}^{q\left(x\right)}}{q\left(x\right)}}\right\}}_n$ is uniformly integrable over $\Omega$. For the rest, following the same arguments as developed in the proof of Lemma 2-(ii) shows that the family ${\left\{{\displaystyle \frac{|\nabla (u_n-u){|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|\nabla (u_n-u){|}^{q\left(x\right)}}{q\left(x\right)}}\right\}}_n$ is also uniformly integrable over $\Omega$. Lastly, using the Vitali Convergence Theorem once more, we obtain

$$\underset{n\to \infty }{lim}{\int }_{\Omega }\left({\displaystyle \frac{|\nabla (u_n-u){|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|\nabla (u_n-u){|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx=0,$$

(59)

which implies, by Proposition 5, that $u_n\to u\in W_0^{1,\mathcal{H}}(\Omega )$.

(v) Using $\left(M\right)$ and Remark 1, it reads

$$\begin{array}{cc}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle & ={\mathcal{M}}_{\varrho }\left(u\right){\int }_{\Omega }{\left(\right|\nabla u|}^{p\left(x\right)}+{\mu \left(x\right)|\nabla u|}^{q\left(x\right)})dx\ge {\displaystyle \frac{{\kappa }_1}{{\left(q^{+}\right)}^{{\alpha }_1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{{\alpha }_1{p}^{-}}\hfill \end{array}$$

(60)

or

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{H},0}}}\ge {\displaystyle \frac{{\kappa }_1}{{\left(q^{+}\right)}^{{\alpha }_1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{{\alpha }_1{p}^{-}-1},\hfill \end{array}$$

(61)

which means that ${\displaystyle \frac{\langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{H},0}}}\to \infty$ as ${\parallel u\parallel }_{1,\mathcal{H},0}\to \infty$, so ${\mathcal{K}}^{\prime }$ is coercive.

Moreover, since ${\mathcal{K}}^{\prime }$ is also strictly monotone, ${\mathcal{K}}^{\prime }$ is an injection. According to the Minty–Browder theorem [30], these two properties together imply that ${\mathcal{K}}^{\prime }$ is a surjection. Therefore, ${\mathcal{K}}^{\prime }$ has an inverse mapping ${\left({\mathcal{K}}^{\prime }\right)}^{-1}:W_0^{1,\mathcal{H}}{(\Omega )}^{*}\to W_0^{1,\mathcal{H}}(\Omega )$.

To show that ${\left({\mathcal{K}}^{\prime }\right)}^{-1}$ is continuous, let $\left(u_n^{*}\right),u^{*}\in W_0^{1,\mathcal{H}}{(\Omega )}^{*}$ with $u_n^{*}\to u^{*}$, and let ${\left({\mathcal{K}}^{\prime }\right)}^{-1}\left(u_n^{*}\right)=u_n,{\left({\mathcal{K}}^{\prime }\right)}^{-1}\left(u^{*}\right)=u$. Then, ${\mathcal{K}}^{\prime }\left(u_n\right)=u_n^{*}$ and ${\mathcal{K}}^{\prime }\left(u\right)=u^{*}$, which means, by the coercivity of ${\mathcal{K}}^{\prime }$, that $\left(u_n\right)$ is bounded in $W_0^{1,\mathcal{H}}(\Omega )$. Thus, there exist $\widehat{u}\in W_0^{1,\mathcal{H}}(\Omega )$ and a subsequence, not relabeled, $\left(u_n\right)\subset W_0^{1,\mathcal{H}}(\Omega )$ such that $u_n\rightharpoonup \widehat{u}$ in $W_0^{1,\mathcal{H}}(\Omega )$. However, by the uniqueness of the weak limit, $\widehat{u}=u$ in $W_0^{1,\mathcal{H}}(\Omega )$. Additionally, since $u_n^{*}\to u^{*}$ in $W_0^{1,\mathcal{H}}{(\Omega )}^{*}$, it reads

$$\underset{n\to \infty }{lim}\langle {\mathcal{K}}^{\prime }\left(u_n\right)-{\mathcal{K}}^{\prime }\left(u\right),u_n-u\rangle =\underset{n\to \infty }{lim}\langle u_n^{*}-u^{*},u_n-u\rangle =0.$$

(62)

Considering that ${\mathcal{K}}^{\prime }$ is of type $\left(S_{+}\right)$, we have $u_n\to u$ in $W_0^{1,\mathcal{H}}(\Omega )$. It can be concluded that ${\left({\mathcal{K}}^{\prime }\right)}^{-1}:W_0^{1,\mathcal{H}}{(\Omega )}^{*}\to W_0^{1,\mathcal{H}}(\Omega )$ is continuous.

(vi) Lastly, we shall show that $\mathcal{K}$ is sequentially weakly lower semicontinuous.

Since ${\mathcal{K}}^{\prime }$ is continuously Gâteaux-differentiable, it is enough to show that $\mathcal{K}$ is convex. To show this, we adopt the approach used in [25] (Lemma 3.4), and, for the sake of completeness, we provide some details. Since $\mathcal{K}\left(u\right)={\widehat{\mathcal{M}}}_{\varrho }\left(u\right)$, we first show that $\widehat{\mathcal{M}}$ is convex and increasing over $(0,\infty )$. By Proposition 8, $\widehat{\mathcal{M}}$ is convex if

$$\widehat{\mathcal{M}}((1-ϵ)t+ϵs)<(1-ϵ){\beta }_1+ϵ{\beta }_2,0<ϵ<1,$$

whenever $\widehat{\mathcal{M}}\left(t\right)<{\beta }_1$ and $\widehat{\mathcal{M}}\left(s\right)<{\beta }_2$, for all $t,s,{\beta }_1,{\beta }_2\in (0,\infty )$. Thus, applying $\left(M\right)$, it reads

$$\begin{array}{c}\hfill \widehat{\mathcal{M}}\left(t\right)\le {\displaystyle \frac{{\kappa }_2}{{\alpha }_2}}{t}^{{\alpha }_2}<{\displaystyle \frac{2^{{\alpha }_2-1}{\kappa }_2}{{\alpha }_2}}{t}^{{\alpha }_2}:={\beta }_1\mathrm{and}\widehat{\mathcal{M}}\left(s\right)\le {\displaystyle \frac{{\kappa }_2}{{\alpha }_2}}{s}^{{\alpha }_2}<{\displaystyle \frac{2^{{\alpha }_2-1}{\kappa }_2}{{\alpha }_2}}{s}^{{\alpha }_2}:={\beta }_2.\end{array}$$

(63)

Therefore,

$$\begin{array}{c}\hfill \widehat{\mathcal{M}}((1-ϵ)t+ϵs)\le {\displaystyle \frac{{\kappa }_2}{{\alpha }_2}}{[(1-ϵ)t+ϵs]}^{{\alpha }_2}<(1-ϵ){\displaystyle \frac{2^{{\alpha }_2-1}{\kappa }_2}{{\alpha }_2}}{t}^{{\alpha }_2}+ϵ{\displaystyle \frac{2^{{\alpha }_2-1}{\kappa }_2}{{\alpha }_2}}{s}^{{\alpha }_2},\end{array}$$

(64)

which shows that $\widehat{\mathcal{M}}$ is convex. To show that $\widehat{\mathcal{M}}$ is increasing over $(0,\infty )$, one can just apply the Fundamental Theorem of Calculus and the condition $\left(M\right)$. Note also that, since $\widehat{\mathcal{M}}$ is convex over $(0,\infty )$, it is continuous on $(0,\infty )$. Putting all these together, since $\widehat{\mathcal{M}}$ is convex and increasing on $(0,\infty )$, and the functional ${\varrho }_{\mathcal{H}}$ is convex on $W_0^{1,\mathcal{H}}(\Omega )$, as the composition of these two maps, $\mathcal{K}$ is also convex on $W_0^{1,\mathcal{H}}(\Omega )$. □

Lemma 3. 

Assume that $\left(f_1\right)$ holds. Then, $\mathcal{J}$ is a continuously Gâteaux-differentiable functional whose Gâteaux derivative ${\mathcal{J}}^{\prime }:W_0^{1,\mathcal{H}}(\Omega )\to W_0^{1,\mathcal{H}}{(\Omega )}^{*}$ given by

$$\langle {\mathcal{J}}^{\prime }\left(u\right),\phi \rangle ={\int }_{\Omega }f(x,u)\phi dx,\forall u,\phi \in W_0^{1,\mathcal{H}}(\Omega )$$

(65)

is compact.

Proof. 

To show that $\mathcal{J}$ is continuously Gâteaux-differentiable, one can argue similarly to Lemma 2, part (ii). Hence, we omit this part and continue in showing that ${\mathcal{J}}^{\prime }$ is compact.

Define the operator ${\mathcal{A}}_f:W_0^{1,\mathcal{H}}(\Omega )\to L^{s^{\prime }\left(x\right)}(\Omega )$ by

$${\mathcal{A}}_f\left(u\right):=f(x,u).$$

(66)

With this characterization, the operator ${\mathcal{A}}_f$ is $L^{s^{\prime }\left(x\right)}(\Omega )$-norm bounded. Indeed, let ${\parallel u\parallel }_{1,\mathcal{H},0}\le 1$. Using the embeddings $L^{\mathcal{H}}(\Omega )\hookrightarrow L^{s\left(x\right)}(\Omega )$, and $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\mathcal{H}}(\Omega )$, it reads

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|{\mathcal{A}}_f\left(u\right)\right|}^{s^{\prime }\left(x\right)}dx& ={\int }_{\Omega }{\left|f(x,u)\right|}^{s^{\prime }\left(x\right)}dx\le {\int }_{\Omega }|{\overline{c}}_1+{\overline{c}}_2{\left|u_n\right|}^{s\left(x\right)-1}{|}^{s^{\prime }\left(x\right)}dx\hfill \\ \hfill & \le c_1{\int }_{\Omega }{\left|u\right|}^{s\left(x\right)}dx+c_2|\Omega |\hfill \\ \hfill & \le c_3{\int }_{\Omega }\left({\left|u\right|}^{s\left(x\right)}+{|\nabla u|}^{p\left(x\right)}+\mu \left(x\right){|\nabla u|}^{q\left(x\right)}\right)dx+c_2|\Omega |\hfill \\ \hfill & \le c_4{\parallel u\parallel }_{1,\mathcal{H},0}+c_2|\Omega |,\hfill \end{array}$$

(67)

where $c_1,c_2,c_3,c_4>0$ are some real parameters whose values are independent of u.

Next, let $u_n\to u$ in $W_0^{1,\mathcal{H}}(\Omega )$, and hence $u_n\to u$ in $L^{s\left(x\right)}(\Omega )$. Due to the standard arguments, there exists a subsequence $\left(u_n\right)$, not relabeled, and a function $\omega$ in $L^{s\left(x\right)}(\Omega )$ satisfying

• $u_n\left(x\right)\to u\left(x\right)$ a.e. in $\Omega$;
• $|u_n\left(x\right)|\le \omega \left(x\right)$ a.e. in $\Omega$ and for all n.

By $\left(f_1\right)$, we have

$$f(x,u_n\left(x\right))\to f(x,u\left(x\right))\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

(68)

and

$$\begin{array}{cc}\hfill \left|f\right(x,u_n\left(x\right)\left)\right|& \le {\overline{c}}_1+{\overline{c}}_2{\left|\omega \left(x\right)\right|}^{s\left(x\right)-1}.\hfill \end{array}$$

(69)

Using Young’s inequality, it reads

$$|f(x,u_n\left(x\right))|\le {\overline{c}}_1+{\displaystyle \frac{{\left({\overline{c}}_2\right)}^{s_{+}}}{s^{-}}}+{\displaystyle \frac{s^{+}-1}{s^{-}}}{\left|\omega \left(x\right)\right|}^{s\left(x\right)}.$$

(70)

Then, by the embeddings $L^{\mathcal{H}}(\Omega )\hookrightarrow L^{s\left(x\right)}(\Omega )$ and $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\mathcal{H}}(\Omega )$, the right-hand side of (69) is integrable. Therefore, by (68) and the Lebesgue dominated convergence theorem, we obtain

$$f(x,u_n)\to f(x,u)\mathrm{in}{L}^1(\Omega ),$$

(71)

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }{|{\mathcal{A}}_f\left(u_n\right)-{\mathcal{A}}_f\left(u\right)|}^{s^{\prime }\left(x\right)}dx=0,\end{array}$$

(72)

which, by Proposition 1, implies that ${\mathcal{A}}_f$ is continuous in $L^{s^{\prime }\left(x\right)}(\Omega )$.

Next, consider the compact embedding operator $i:W_0^{1,\mathcal{H}}(\Omega )\to L^{s\left(x\right)}(\Omega )$. Then, the adjoint operator $i^{*}$ of i, given by $i^{*}:L^{s^{\prime }\left(x\right)}(\Omega )\to W_0^{1,\mathcal{H}}{(\Omega )}^{*}$, is also compact. Therefore, if we set ${\mathcal{J}}^{\prime }=i^{*}\circ {\mathcal{A}}_f$, then ${\mathcal{J}}^{\prime }:W_0^{1,\mathcal{H}}(\Omega )\to W_0^{1,\mathcal{H}}{(\Omega )}^{*}$ is compact. □

We are ready to present the proof of the main result.

Proof. 

(Proof of Theorem 1)

The required regularity conditions for $\mathcal{K}$ and $\mathcal{J}$ are proven in Lemmas 2 and 3.

Moreover, it is clear that

$$\underset{u\in W_0^{1,\mathcal{H}}}{inf}\mathcal{K}\left(u\right)=\mathcal{K}\left(0\right)=\mathcal{J}\left(0\right)=0.$$

(73)

Note that, if we define

$$\delta \left(x\right)=sup\{\delta >0:B(x,\delta )\subseteq \Omega \},\forall x\in \Omega ,$$

(74)

and consider that $\Omega$ is open and connected in ${\mathbb{R}}^N$, it can easily be shown that there exists $x_0\in \Omega$ such that $B(x_0,R)\subseteq \Omega$ with $R={sup}_{x\in \Omega }\delta \left(x\right)$.

First, we define the cut-off function $\overline{u}\in W_0^{1,\mathcal{H}}(\Omega )$ by the formula

$$\overline{u}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}x\in \Omega \setminus B(x_0,R),\hfill \\ \delta ,\hfill & \mathrm{if}x\in B(x_0,R/2),\hfill \\ {\displaystyle \frac{2\delta }{R}}(R-|x-x_0\left|\right),\hfill & \mathrm{if}x\in B(x_0,R)\setminus B(x_0,R/2):=\widehat{B}.\hfill \end{array}\right.$$

(75)

Then, using $\left(M\right)$, we have

$$\begin{array}{cc}\hfill \mathcal{K}\left(\overline{u}\right)& ={\int }_0^{{\varrho }_{\mathcal{H}}\left(\overline{u}\right)}\mathcal{M}\left(s\right)ds\ge {\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\left({\int }_{\widehat{B}}\left(|\nabla \overline{u}{|}^{p\left(x\right)}+\mu \left(x\right){|\nabla \overline{u}|}^{q\left(x\right)}\right)dx\right)}^{{\alpha }_1}\hfill \\ \hfill & \ge {\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{{\alpha }_1(p^{-}\wedge p^{+})}{\omega }_N^{{\alpha }_1}{\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)}^{{\alpha }_1}\hfill \\ \hfill & ={\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\omega }_N^{{\alpha }_1}{R}^{N{\alpha }_1}{(2^N-1)}^{{\alpha }_1}{2}^{-N{\alpha }_1}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{{\alpha }_1(p^{-}\wedge p^{+})},\hfill \end{array}$$

(76)

and, similarly,

$$\begin{array}{cc}\hfill \mathcal{K}\left(\overline{u}\right)& \le {\displaystyle \frac{{\kappa }_2}{{\alpha }_2{\left(p^{-}\right)}^{{\alpha }_2}}}{\left({\int }_{\widehat{B}}\left(|\nabla \overline{u}{|}^{p\left(x\right)}+\mu \left(x\right){|\nabla \overline{u}|}^{q\left(x\right)}\right)dx\right)}^{{\alpha }_2}\hfill \\ \hfill & \le {\displaystyle \frac{{\kappa }_2{(1+|\mu |}_{\infty }{)}^{{\alpha }_2}}{{\alpha }_2{\left(p^{-}\right)}^{{\alpha }_2}}}{\omega }_N^{{\alpha }_2}{\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)}^{{\alpha }_2}{\left({\displaystyle \frac{2\delta }{R}}\right)}^{{\alpha }_2(p^{-}\vee q^{+})},\hfill \end{array}$$

(77)

where ${\omega }_N:={\displaystyle \frac{{\pi }^{N/2}}{N/2\Gamma (N/2)}}$ is the volume of the unit ball in ${\mathbb{R}}^N$, where $\Gamma$ is the Gamma function.

By $\left(f_3\right)$, it reads

$$\begin{array}{cc}\hfill \mathcal{J}\left(\overline{u}\right)& ={\int }_{B(x_0,R/2)}F(x,\overline{u})dx\ge \underset{x\in \Omega }{inf}F(x,\delta ){\omega }_N{\left({\displaystyle \frac{R}{2}}\right)}^N.\hfill \end{array}$$

(78)

Therefore,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathcal{J}\left(\overline{u}\right)}{\mathcal{K}\left(\overline{u}\right)}}& \ge {\displaystyle \frac{2^{N({\alpha }_2-1)}{\alpha }_2{\left(p^{-}\right)}^{{\alpha }_2}{inf}_{x\in \Omega }F(x,\delta )}{{\kappa }_2{(1+|\mu |}_{\infty }{)}^{{\alpha }_2}{\omega }_N^{{\alpha }_2-1}{(2^N-1)}^{{\alpha }_2}{R}^{N({\alpha }_2-1)}{\left(\frac{2\delta }{R}\right)}^{{\alpha }_2(p^{-}\vee q^{+})}}}.\hfill \end{array}$$

(79)

Note that, for any $u\in {\mathcal{K}}^{-1}\left((-\infty ,r]\right)$, by $\left(M\right)$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{{\kappa }_1}{{\alpha }_1}}{\varrho }_{\mathcal{H}}{\left(u\right)}^{{\alpha }_1}\le \mathcal{K}\left(u\right)\le r.\end{array}$$

(80)

Thus, by the involved embeddings, it reads

$$\begin{array}{cc}\hfill \mathcal{J}\left(u\right)& \le {\int }_{\Omega }{\overline{c}}_1\left|u\right|dx+{\int }_{\Omega }{\overline{c}}_2{\left|u\right|}^{s\left(x\right)}dx\le {\overline{c}}_1{c}_1{\left(\mathcal{H}\right)\parallel u\parallel }_{1,\mathcal{H},0}+{\overline{c}}_2{c}_2{\left(\mathcal{H}\right)}^{s^{+}}{\parallel u\parallel }_{1,\mathcal{H},0}^{s^{+}}\hfill \\ \hfill & \le {\overline{c}}_1{c}_{\mathcal{H}}{\left(q^{+}\right)}^{{\displaystyle \frac{1}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}+{\overline{c}}_2{c}_{\mathcal{H}}^{s^{+}}{\left(q^{+}\right)}^{{\displaystyle \frac{s^{+}}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}},\hfill \end{array}$$

(81)

and, hence,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{r}}\mathcal{J}\left(u\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{r}}\{{\overline{c}}_1{c}_{\mathcal{H}}{\left(q^{+}\right)}^{{\displaystyle \frac{1}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{1}{{\alpha }_1(p^{-}\wedge q^{+})}}}+{\overline{c}}_2{c}_{\mathcal{H}}^{s^{+}}{\left(q^{+}\right)}^{{\displaystyle \frac{s^{+}}{p^{-}}}}{\left({\displaystyle \frac{{\alpha }_1}{{\kappa }_1}}\right)}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}}{r}^{{\displaystyle \frac{s^{-}\wedge s^{+}}{{\alpha }_1(p^{-}\wedge q^{+})}}}\},\hfill \end{array}$$

(82)

where $c_{\mathcal{H}}:=max\{c_1\left(\mathcal{H}\right),c_2\left(\mathcal{H}\right)\}$ and $c_1\left(\mathcal{H}\right),c_2\left(\mathcal{H}\right)$ are the best embedding constants. Then, using $\left(f_4\right)$ and (79) provides

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{r}}\underset{\mathcal{K}\left(u\right)\le r}{sup}\mathcal{J}\left(u\right)<{\displaystyle \frac{\mathcal{J}\left(\overline{u}\right)}{\mathcal{K}\left(\overline{u}\right)}}.\end{array}$$

(83)

Thus, condition $\left(a1\right)$ of Lemma 1 is verified.

Next, we show that condition $\left(a2\right)$ of Lemma 1 is satisfied—that is, the functional ${\mathcal{I}}_{\lambda }(·):=\mathcal{K}(·)-\lambda \mathcal{J}(·)$ is coercive.

Using the embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{r\left(x\right)}(\Omega )$, and Lemma 2-(i), it reads

$${\mathcal{I}}_{\lambda }\left(u\right)\ge {\displaystyle \frac{{\kappa }_1}{{\alpha }_1{\left(q^{+}\right)}^{{\alpha }_1}}}{\parallel u\parallel }_{1,\mathcal{H},0}^{{\alpha }_1{p}^{-}}-\lambda {\overline{c}}_3(c_{\mathcal{H}}^{r^{+}}{\parallel u\parallel }_{1,\mathcal{H},0}^{r^{+}}+|\Omega |),$$

(84)

which implies that, for any $\lambda >0$, ${\mathcal{I}}_{\lambda }$ is coercive.

In conclusion, by Lemma 1, for any $\lambda \in {\lambda }_{r,\delta }\subseteq \left({\displaystyle \frac{\mathcal{K}\left(\overline{u}\right)}{\mathcal{J}\left(\overline{u}\right)}},{\displaystyle \frac{r}{{sup}_{\mathcal{K}\left(u\right)\le r}\mathcal{J}\left(u\right)}}\right)$, problem ($𝒫$) admits at least three distinct weak solutions. □

4. Conclusions

In this work, we have investigated a class of double-phase variable-exponent Kirchhoff problems and established the existence of at least three distinct weak solutions. The problem framework extends classical Kirchhoff-type equations by incorporating a double-phase operator with variable growth, thereby capturing anisotropic and heterogeneous diffusion phenomena within a unified variational setting. A key contribution of this paper lies in the rigorous analysis of the associated energy functional, for which we have proven essential regularity properties such as $C^1$-smoothness, the $\left(S_{+}\right)$-condition, and sequential weak lower semicontinuity. These results enable the application of a three critical point theorem due to Bonanno and Marano, ensuring the existence of multiple solutions under suitable assumptions. As a novel tool in the analysis, we introduce an n-dimensional vector inequality (Proposition 9), which can be utilized to handle the technical challenges posed by variable-exponent, nonstandard growth functionals. This auxiliary result plays a fundamental role in proving the differentiability and compactness properties of the functional. Our findings extend the theoretical framework for Kirchhoff-type problems with a double-phase structure. Future work may explore further generalizations, including nonlocal operators, critical growth terms, or imposing singular perturbations in the same setting.