Existence and Uniqueness Results for a Kirchhoff Double-Phase Problem Involving the ψ-Hilfer Derivative

Abstract

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems involving variable-exponent growth conditions. The first problem analyzes general nonlinear sources and formulates the solution as a fixed point of a nonlinear operator. Precisely, by proving that the functional energy is coercive, hemicontinuous, and strictly monotone, we establish the existence and the uniqueness of weak solutions via monotone operator theory. The second problem incorporates a convection-type nonlinearity, which breaks variational structure and requires the more robust theory of pseudomonotone operators. Under suitable growth and mixed-order assumptions on the nonlinearity, we prove the existence of at least one weak solution. The main tools are grounded in variable-exponent Lebesgue and Musielak–Orlicz–Sobolev spaces, with compact embeddings, modular estimates, and fractional integral identities playing a key role in the proofs. We note that the results contribute to the mathematical modeling of phenomena involving nonlocal elasticity, viscoelastic materials, phase-transition media, and fractional dynamical systems where the stiffness of the medium depends on the total deformation (Kirchhoff effect) and the energy density alternates between distinct growth regimes (double-phase). The $\psi$-Hilfer derivative enhances the scope by enabling models with tunable memory and hereditary effects.

1. Introduction

Fractional calculus generalizes classical calculus by allowing derivatives and integrals of arbitrary non-integer order. It presents a more precise mathematical framework for memory systems with long-term dependencies. Several physical processes, such as viscoelastic materials and anomalous diffusion, are better described by fractional models (see [1]). In control theory, fractional operators improve system stability and robustness (see [2]). Fractional calculus also plays a key role in modeling complex electrical circuits and signal processing systems (see [3]). In biology, it allows the description of neuronal dynamics and population growth with memory effects (see [4]). Financial models also use fractional calculus to understand market disturbances and long-term correlations (see [5]). For interested readers, we can cite other models in many phenomena of science and engineering, for example, the works of [6] (application in mechanics), [7] (application in dynamical systems, and [8] (application in modeling blood alcohol concentration). So, we can conclude that fractional operators allow the link between theory and real-world phenomena and offer a more nuanced understanding and greater flexibility than classical mathematical tools.

Given its importance, several researchers have recently tried to develop projects around this topic.

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

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill -\mathcal{M}{\left({\xi }_{\mathcal{K}}\left(u\right)\right)}^{\mathrm{H}}{\mathbb{D}}_T^{\alpha ,\nu ;\psi }\left({|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u+\mu \left(x\right){{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)& =f\mathrm{in}\mathrm{\Omega },\hfill \\ \hfill u& =0\mathrm{on}\mathsf{\partial }\mathrm{\Omega },\hfill \end{array}\hfill \end{array}\right.$$

(1)

where $\mu (·)\ge 0$, $\mathrm{\Omega }:=(0,T)\times (0,T)\times (0,T)$, ${}^{\mathrm{H}}\mathbb{D}_T^{\alpha ,\nu ;\psi }(·)$ and ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(·)$ are the $\psi$-Hilfer fractional derivatives ($\psi$-HFD) of order ${\displaystyle \frac{1}{p^{+}}}<\alpha <1$ and of type $0\le \nu \le 1$. $\mathcal{M}$ is a $C^1$-continuous nondecreasing function; $p,q\in C_{+}\left(\overline{\mathrm{\Omega }}\right)$, f is a nontrivial function in the first main result, and $f:\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^3\to \mathbb{R}$ is a Carathéodory function in the second main result.

$$\begin{array}{c}\hfill {\xi }_{\mathcal{K}}\left(u\right):={\int }_{\mathrm{\Omega }}\left({\displaystyle \frac{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx.\end{array}$$

(2)

Many researchers have concentrated on developing new fractional operators like the $\psi$-Hilfer fractional derivative, for instance, see the papers of Sahbani et al. [9], Vanterler et al. [10,11,12], and the references therein.

The first important point in our study is that we consider a $\psi$-Hilfer fractional derivative and a double-phase operator (introduced by Zhikov, see [13,14]), which exhibits behavior that alternates between two distinct elliptic scenarios. In the last few years, there have been several papers dealing with double-phase operators; for example, we refer to the papers [11,15,16,17,18,19,20,21,22,23,24].

The second important point in our study is that we consider a Kirchhoff problem which was introduced by Kirchhoff [25]; it is noted that Kirchhoff-type problems generally refer to mathematical models related to the analysis of electrical circuits using Kirchhoff’s laws, which are, in fact, fundamental principles in circuit theory; we refer the interested readers to the papers [9,10,26] and the references therein. More precisely, Sousa [26] considered the double-phase fractional Kirchhoff-type equation

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill -M{\left({\xi }_{\mathcal{K}}\left(u\right)\right)}^{\mathrm{H}}{\mathbb{D}}_T^{\alpha ,\nu ;\psi }\left({|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u+\mu \left(x\right){{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2} {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)& =f(x,u)\mathrm{in}\mathrm{\Omega },\hfill \\ \hfill u& =0\mathrm{on}\mathsf{\partial }\mathrm{\Omega },\hfill \end{array}\hfill \end{array}\right.$$

(3)

where $\mathrm{\Omega }:=(0,T)\times (0,T)$, ${\xi }_{\mathcal{K}}$ is defined in (2), ${}^{\mathrm{H}}\mathbb{D}_T^{\alpha ,\nu ;\psi }(·)$, and ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(·)$ are the $\psi$-HFD of order ${\displaystyle \frac{1}{p^{+}}}<\alpha <1$ and of type $0\le \nu \le 1$. $p,q\in C_{+}\left(\overline{\mathrm{\Omega }}\right)$ with $1<p\left(x\right)<q\left(x\right)$ and $0\le \mu (·)\in L^{\infty }\left(\mathrm{\Omega }\right)$. By using the variational method and the Kajikiya theorem and taking into consideration some suitable assumptions on the functions M and f, the authors proved the existence and multiplicity of solutions of problem (3).

The third important point in our work is that we consider a nonlinearity f containing a derivative. Such functions are usually called convection terms. The difficulty with the gradient-dependent term is the nonvariational character of the problem. Nevertheless, there exist several papers concerning existence and multiplicity results. For other existence results on quasilinear equations with convection term and the p-Laplace or the $(p,q)$-Laplace differential operator, we refer to the papers of Bai-Gasiňski-Papageorgiou [27], M. Colombo, G. Mingione [21], M. Colombo, G. Mingione [22], Dupaigne-Ghergu-Rǎdulescu [28], Faraci-Motreanu-Puglisi [29], Faria-Miyagaki-Motreanu [30], and Tanaka [31] and the references therein. More precisely, Gasiński et al. [32] considered the double-phase problems with convection term:

$$\left\{\begin{array}{c}-div\left({|\nabla v|}^{p_1-2}\nabla v+\mu \left(x\right){|\nabla v|}^{p_2-2}\nabla v\right)=f(\xi ,v,\nabla v),\mathrm{in}\mathrm{\Lambda },\hfill \\ \\ v=0,\mathrm{on}\mathsf{\partial }\mathrm{\Lambda },\hfill \end{array}\right.$$

where $\mathrm{\Lambda }\subset {\mathbf{R}}^N$, $N\ge 2$ is a bounded domain with Lipschitz boundary $\mathsf{\partial }\mathrm{\Lambda }$, $1<p_1<p_2<N$ and $\mu :\overline{\mathrm{\Lambda }}\to [0,\infty )$ is supposed to be Lipschitz continuous. Under quite general assumptions on the convection term, the authors proved the existence of a weak solution by applying the theory of pseudomonotone operators.

Based on the research papers mentioned above, we use the theory of pseudomonotone operators to study a double-phase problem with convection and Kirchhoff terms involving the $\psi$-Hilfer fractional derivative and variable exponents. More precisely, in Section 3, we prove the existence of a unique solution for problem (1). In Section 4, we prove the existence of at least one solution for such a problem. We note that the studied problem in this paper represents a significant advancement in the field of fractional calculus and nonlinear operators. The following points highlight how this work compares to the existing literature:

• Generalization of Fractional Operators: While many researchers have focused on developing new fractional operators [9,10,11,12], this work specifically utilizes the $\psi$-Hilfer fractional derivative to enhance models with tunable memory and hereditary effects [33,34,35].
• Double-Phase Behavior: Unlike standard elliptic problems, the operator studied here (originally introduced by Zhikov [13,14]) exhibits double-phase behavior that alternates between two distinct elliptic scenarios [26]. While several recent papers have dealt with double-phase operators [16,17,18,19,20,21,22,23,24], this research is among the first to address them simultaneously with $\psi$-Hilfer derivatives and Kirchhoff-type terms [9,26].
• Kirchhoff-Type Problems: Kirchhoff’s problem, originally introduced for electrical circuit analysis and nonlocal elasticity [25], has been extended by authors like Sousa [26] to include double-phase fractional equations [36]. This paper builds upon those foundations by incorporating variable-exponent growth conditions [9,34].
• Handling Nonvariational Structures: A major challenge in the literature is the presence of gradient-dependent terms (convection terms), which break the variational structure of the problem [31,37]. While previous works on quasilinear equations with convection terms used p-Laplace or $(p,q)$-Laplace operators [6,24,38,39], this work applies the theory of pseudomonotone operators to establish existence results for a much more generalized Kirchhoff double-phase model [20,35].

• In conclusion, the novelty of this work lies in the interaction between the $\psi$-Hilfer fractional operator and the double-phase operator under variable exponent growth conditions. While these components have been studied individually, their coupling creates new challenges in the modular estimates and the compactness of embeddings in Musielak–Orlicz–Sobolev spaces. Also, the studied problem is related to fluid flow in a heterogeneous porous medium, specifically focusing on groundwater flow, where,

•

$u\left(x\right)$: hydraulic head (pressure).

•

${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\left(x\right)$: hydraulic gradient, which determines the direction and speed of groundwater flow.

•

$\mu (·)\in L^{\infty }\left(\mathrm{\Omega }\right)$ such that $\mu \left(x\right)\ge 0$ for all $x\in \overline{\mathrm{\Omega }}$: heterogeneity factor accounting for permeability variations.

•

The operator

$${\mathcal{L}}_1\left(u\right):{=}^{\mathrm{H}}{\mathbb{D}}_T^{\alpha ,\nu ;\psi }\left({|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u+\mu \left(x\right){{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right),$$

represents two soil phases: clay and sand.

2. Preliminaries

In this section, we present some definitions and properties of the variable exponent Lebesgue space $L^{p\left(x\right)}\left(\mathrm{\Omega }\right)$, the $\psi$-fractional Sobolev space ${\mathcal{H}}_{p\left(x\right)}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$, the Musielak–Orlicz space $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$, and the Musielak–Orlicz–Sobolev space ${\mathcal{H}}_{\mathcal{K}}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$. For more details, see the references [16,17,18,19,20,21,22,23,24,32,36].

$${\mathcal{C}}_{+}\left(\overline{\mathrm{\Omega }}\right)=\left\{p\in C\left(\overline{\mathrm{\Omega }}\right),p\left(x\right)>1\mathrm{for}\mathrm{all}x\in \overline{\mathrm{\Omega }}\right\}.$$

For any $p\in {\mathcal{C}}_{+}\left(\overline{\mathrm{\Omega }}\right)$, we denote by $M\left(\mathrm{\Omega }\right)$ the space of all measurable functions $u:\mathrm{\Omega }\to \mathbb{R}$, and we introduce the variable exponent Lebesgue space as:

$$L^{p\left(x\right)}\left(\mathrm{\Omega }\right)=\left\{u\in M\left(\mathrm{\Omega }\right),{\int }_{\mathrm{\Omega }}{\left|u\left(x\right)\right|}^{p\left(x\right)}dx<\infty \right\}.$$

which is endowed with the following norm:

$${\left|u\right|}_{p\left(x\right)}=inf\left\{\xi >0:{\int }_{\mathrm{\Omega }}{\left|{\displaystyle \frac{u\left(x\right)}{\xi }}\right|}^{p\left(x\right)}dx\le 1\right\}.$$

It is noted that the $\left(L^{p\left(x\right)}\left(\mathrm{\Omega }\right),{\left|u\right|}_{p\left(x\right)}\right)$ is a Banach space.

Next, we define the modular $\rho :L^{p\left(x\right)}\left(\mathrm{\Omega }\right)\to \mathbb{R}$, by

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

Some important relationships between the modular and the norm are given in the following propositions.

Proposition 1

([9,36]). If $u,u_n\in L^{p\left(x\right)}\left(\mathrm{\Omega }\right)$, we get

$\left(i\right)$

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

$\left(ii\right)$

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

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

$\left(iii\right)$

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

Proposition 2

([9,36]). Let $p_1$ be a measurable function in $p_1\in L^{\infty }\left(\mathrm{\Omega }\right)$ and $p_2$ is a measurable function such that

$$1\le p_1\left(x\right)p_2\left(x\right)\le \infty ,\mathit{for}a.e.x\in \mathrm{\Omega }.$$

Let $u\in L^{p_2\left(x\right)}\left(\mathrm{\Omega }\right),$ with $u\ne 0$. Then, we have

$\left(i\right)$

${\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}\le 1⟹{\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}^{p_1^{+}}\le {\left|{\left|u\right|}^{p_1\left(x\right)}\right|}_{p_2\left(x\right)}\le {\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}^{p_1^{-}}$

$\left(ii\right)$

${\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}\ge 1⟹{\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}^{p_1^{-}}\le {\left|{\left|u\right|}^{p_1\left(x\right)}\right|}_{p_2\left(x\right)}\le {\left|u\right|}_{p_1\left(x\right)p_2\left(x\right)}^{p_1^{+}}$

$\left(iii\right)$

If $p_1\left(x\right)=p$ is a constant, then

$${\left|{\left|u\right|}^p\right|}_{p_2\left(x\right)}={\left|u\right|}_{p{p}_2\left(x\right)}^p.$$

The $\psi$-fractional Sobolev space (see [10,11,12]) is given by

$$\begin{array}{c}\hfill {\mathcal{H}}_{p\left(x\right)}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)=\left\{u\in L^{p\left(x\right)}\left(\mathrm{\Omega }\right):\left|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right|\in L^{p\left(x\right)}\left(\mathrm{\Omega }\right)\right\}\end{array}$$

with the norm

$$\begin{array}{c}\hfill {\parallel u\parallel }_{1,p\left(x\right)}=\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{p\left(x\right)}+{\parallel u\parallel }_{p\left(x\right)}.\end{array}$$

Let ${\mathcal{H}}_{p\left(x\right),0}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$ be the closure of $C_0^{\infty }\left(\mathrm{\Omega }\right)$ in ${\mathcal{H}}_{p\left(x\right)}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$, equipped with the following equivalent norm

$$\begin{array}{c}\hfill {\parallel u\parallel }_{p\left(x\right),0}={\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\parallel }_{p\left(x\right)},\end{array}$$

Hereafter, we fix $p\in {\mathcal{C}}_{+}\left(\overline{\mathrm{\Omega }}\right)$ such that

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

(4)

Proposition 3

([10,11,12,36]). Assume that Equation (4) holds, then the spaces $L^{p\left(x\right)}\left(\mathrm{\Omega }\right)$, ${\mathcal{H}}_{p\left(x\right)}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$, and ${\mathcal{H}}_{p\left(x\right),0}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$ are reflexive.

The following embeddings result plays a very important role in this paper.

Proposition 4

([10,11,12]). Let $m\in C\left(\overline{\mathrm{\Omega }}\right)$ be such that

$$1\le m\left(x\right)<p_{\alpha }^{*}\left(x\right),\mathit{for}\mathit{all}x\in \overline{\mathrm{\Omega }}.$$

Then, the embeddings ${\mathcal{H}}_{p\left(x\right)}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)\hookrightarrow L^{m\left(x\right)}\left(\mathrm{\Omega }\right)$ and ${\mathcal{H}}_{p\left(x\right),0}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)\hookrightarrow L^{m\left(x\right)}\left(\mathrm{\Omega }\right)$ are compact and continuous, where

$$p_{\alpha }^{*}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3p\left(x\right)}{3-\alpha p\left(x\right)}}& \mathit{if}\alpha p\left(x\right)<3,\\ +\infty & \mathit{if}\alpha p\left(x\right)\ge 3.\end{array}\right.$$

Now, we define the double-phase operator (see [11,16,17]) by

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

The modular ${\rho }_{\mathcal{K}}(·)$ is defined by

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

The Musielak–Orlicz space $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$, is defined by

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

endowed with the Luxemburg norm

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

The Musielak–Orlicz–Sobolev space ${\mathcal{H}}_{\mathcal{K}}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$ is defined by

$${\mathcal{H}}_{\mathcal{K}}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)=\left\{u\in L^{\mathcal{K}}\left(\mathrm{\Omega }\right):\left|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right|\in L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\right\},$$

and equipped with the norm

$${\parallel u\parallel }_{1,\mathcal{K}}:=\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}+{\parallel u\parallel }_{\mathcal{K}}.$$

The closure of the space $\overline{C_0^{\infty }\left(\mathrm{\Omega }\right)}$ in ${\mathcal{H}}_{\mathcal{K}}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$ will be denoted by ${\mathcal{H}}_{\mathcal{K},0}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$.

Proposition 5

([11,16,17]). If $\psi \in L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$. Then

$\left(i\right)$

$\psi \ne 0$, then ${\parallel \psi \parallel }_{\mathcal{K}}=\zeta \iff {\rho }_{\mathcal{K}}\left({\displaystyle \frac{\psi }{\zeta }}\right)=1$.

$\left(ii\right)$

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

$\left(iii\right)$

${\parallel \psi \parallel }_{\mathcal{K}}<1\Rightarrow {\parallel \psi \parallel }_{\mathcal{K}}^{r^{+}}\le {\rho }_{\mathcal{K}}\left(\psi \right)\le {\parallel \psi \parallel }_{\mathcal{K}}^{p^{-}}$.

$\left(iv\right)$

${\parallel \psi \parallel }_{\mathcal{K}}>1\Rightarrow {\parallel \psi \parallel }_{\mathcal{K}}^{p^{-}}\le {\rho }_{\mathcal{K}}\left(\psi \right)\le {\parallel \psi \parallel }_{\mathcal{K}}^{r^{+}}$.

$\left(v\right)$

${\parallel \psi \parallel }_{\mathcal{K}}\to 0\iff {\rho }_{\mathcal{K}}\left(\psi \right)\to 0$.

$\left(vi\right)$

${\parallel \psi \parallel }_{\mathcal{K}}\to +\infty \iff {\rho }_{\mathcal{K}}\left(\psi \right)\to +\infty$.

$\left(vii\right)$

${\parallel \psi \parallel }_{\mathcal{K}}\to 1\iff {\rho }_{\mathcal{K}}\left(\psi \right)\to 1$.

$\left(viii\right)$

If ${\psi }_n\to \psi$ in $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$, then ${\rho }_{\mathcal{K}}\left({\psi }_n\right)\to {\rho }_{\mathcal{K}}\left(\psi \right)$.

Next, we assume the following hypotheses:

$\left(H_1\right)$

$p,q\in C_{+}\left(\overline{\mathrm{\Omega }}\right)$ with $2<p^{-}\le p\left(x\right)<q\left(x\right)<p_{\alpha }^{*}\left(x\right)$.

$\left(H_2\right)$

$\mu (·)\in L^{\infty }\left(\mathrm{\Omega }\right)$ such that $\mu \left(x\right)\ge 0$ for all $x\in \overline{\mathrm{\Omega }}$.

$\left(H_3\right)$

$\mathcal{M}:[0,\infty )\to [m_0,\infty )$ is a $C^1$-continuous nondecreasing function, for some positive constant $m_0.$

Proposition 6

([10,11]). Under hypotheses $\left(H_1\right)$ and $\left(H_2\right)$, $L^{\mathcal{K}}\left(\mathrm{\Omega }\right),{\mathcal{H}}_{\mathcal{K}}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$, and X are reflexive Banach spaces.

Proposition 7

([10,11]). Under hypotheses $\left(H_1\right)$ and $\left(H_2\right)$, the embedding

$$\begin{array}{c}\hfill X\hookrightarrow L^{r\left(\xi \right)}\left(\mathrm{\Omega }\right)\end{array}$$

(5)

is compact whenever $r\left(\xi \right)<p_{\alpha }^{*}\left(\xi \right)$.

Remark 1.

The embedding $X\hookrightarrow L^{p(·)}\left(\mathrm{\Omega }\right)$ is continuous, and

$${\parallel u\parallel }_{p(·)}\le c_{\mathcal{K}}{\parallel u\parallel }_{1,\mathcal{K},0}.$$

where $c_{\mathcal{K}}$ is a positive constant. Then, the embedding $X\hookrightarrow L^{\mathcal{H}}\left(\mathrm{\Omega }\right)$ is compact. So, X can be equipped with the norm

$${\parallel u\parallel }_{1,\mathcal{K},0}:={\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\parallel }_{\mathcal{K}}.$$

Also, we introduce the spaces

$$L_{\mu }^{p(·)}\left(\mathrm{\Omega }\right)=\left\{u\in M\left(\mathrm{\Omega }\right);{\int }_{\mathrm{\Omega }}\mu \left(x\right){\left|u\right|}^{p\left(x\right)}dx<+\infty \right\},$$

equipped with with the seminorms

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

So, the embedding $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\hookrightarrow L_{\mu }^{p(·)}\left(\mathrm{\Omega }\right)$ is continuous.

Finally, we introduce the functional:

$${\xi }_{\mathcal{K}}\left(u\right)={\int }_{\mathrm{\Omega }}\left({\displaystyle \frac{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)}}{p\left(x\right)}}+\mu \left(x\right){\displaystyle \frac{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{q\left(x\right)}}{q\left(x\right)}}\right)dx.$$

The following proposition gives some properties of the functional ${\xi }_{\mathcal{K}}$.

Proposition 8

([11]). We have:

$\left(i\right)$

${\xi }_{\mathcal{K}}\in C^1(X,\mathbb{R})$ and

$$\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),\phi \rangle ={\int }_{\mathrm{\Omega }}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u)·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi dx,$$

for all $u,\phi \in X$.

$\left(ii\right)$

${\xi }_{\mathcal{K}}^{\prime }$$\in \left(S_{+}\right)$, i.e.,

$$u_n\rightharpoonup u\mathit{in}X$$

(6)

and

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

(7)

then

$$u_n\to u\mathit{in}X.$$

(8)

An important relationship between the modular ${\rho }_{\mathcal{K}}$, the norm ${\parallel u\parallel }_{1,\mathcal{K},0}$, and the functional ${\xi }_{\mathcal{K}}$ is as follows:

Lemma 1.

We have the relations:

$${\xi }_{\mathcal{K}}\left(u\right)\le {\displaystyle \frac{1}{p^{-}}}{\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\le {\displaystyle \frac{1}{p^{-}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{q^{+}}\mathit{if}{\parallel u\parallel }_{1,\mathcal{K},0}>1,$$

$${\xi }_{\mathcal{K}}\left(u\right)\ge {\displaystyle \frac{1}{q^{+}}}{\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\ge {\displaystyle \frac{1}{q^{+}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{p^{-}}\mathit{if}{\parallel u\parallel }_{1,\mathcal{K},0}>1,$$

$${\xi }_{\mathcal{K}}\left(u\right)\le {\displaystyle \frac{1}{p^{-}}}{\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\le {\displaystyle \frac{1}{p^{-}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{p^{-}}\mathit{if}{\parallel u\parallel }_{1,\mathcal{K},0}\le 1.$$

$${\xi }_{\mathcal{K}}\left(u\right)\ge {\displaystyle \frac{1}{q^{+}}}{\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\ge {\displaystyle \frac{1}{q^{+}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{q^{+}}\mathit{if}{\parallel u\parallel }_{1,\mathcal{K},0}\le 1.$$

$$0<\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),u\rangle ={\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right),$$

$$0<p^{-}{\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\le \langle {\rho }_{\mathcal{K}}^{\prime }\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right),{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\rangle .$$

Finally, the following technical lemma will be useful in the sequel.

Lemma 2.

Let $u,v\in {\mathbb{R}}^N$, for any $1\le q<\infty$ and $t_1,t_2>0$, we have

$$\left|t_1{\left|u\right|}^{q-2}u-t_2{\left|v\right|}^{q-2}v\right|\le \left(t_1+|t_1-t_2|\right)\left|{\left|u\right|}^{q-2}u-{\left|v\right|}^{q-2}v\right| + |t_1-t_2{\left|\right|u|}^{q-1}.$$

Proof. 

It is clear that if $t_1=t_2$ or ${\left|u\right|}^{q-2}u={\left|v\right|}^{q-2}v$, then the inequality holds. Next, we assome that $t_1\ne t_2$ and ${\left|u\right|}^{q-2}u\ne {\left|v\right|}^{q-2}v$.

• Put

$$g(u,v)={\displaystyle \frac{|t_1{\left|u\right|}^{q-2}u-t_2{\left|v\right|}^{q-2}v|}{{\left|\right|u|}^{q-2}u-{\left|v\right|}^{q-2}v|}}.$$

Remark that g is invariant by any orthogonal transformation F; that is, $g(Fu,Fv)=g(u,v)$ for all $u,v\in {\mathbb{R}}^N$. Thus, using this argument and the homogeneity of g, we can let $u=\left|u\right|e_1$ and assume that $u=e_1$. Then

$$g(e_1,v)={\displaystyle \frac{|t_1{e}_1-t_2{\left|v\right|}^{q-2}v|}{|e_1-{\left|v\right|}^{q-2}v|}}.$$

(9)

We have

$$\begin{array}{cc}\hfill |t_1{e}_1-t_2{\left|v\right|}^{q-2}v|& =t_1|e_1-{\displaystyle \frac{t_2}{t_1}}{\left|v\right|}^{q-2}v|=t_1|(e_1-{\left|v\right|}^{q-2}v)+\left(1-{\displaystyle \frac{t_2}{t_1}}\right){\left|v\right|}^{q-2}v|\hfill \\ \hfill & \le t_1|e_1-{\left|v\right|}^{q-2}v| + |t_1-t_2{\left|\right|\left|v\right|}^{q-2}v|\hfill \\ \hfill & \le t_1|e_1-{\left|v\right|}^{q-2}v| + |t_1-t_2|\left(|e_1-{\left|v\right|}^{q-2}v| + |e_1|\right)\hfill \\ \hfill & \le t_1|e_1-{\left|v\right|}^{q-2}v| + |t_1-t_2\left|\right|e_1-{\left|v\right|}^{q-2}v| + |t_1-t_2|\hfill \\ \hfill & \le |e_1-{\left|v\right|}^{q-2}v|\left(t_1+|t_1-t_2|\right)+|t_1-t_2|.\hfill \end{array}$$

(10)

So, by (9), we get

$$\begin{array}{cc}\hfill g(e_1,v)& \le {\displaystyle \frac{|e_1-{\left|v\right|}^{q-2}v|\left(t_1+|t_1-t_2|\right)+|t_1-t_2|}{|e_1-{\left|v\right|}^{q-2}v|}}\hfill \\ \hfill & \le t_1+|t_1-t_2|+{\displaystyle \frac{|t_1-t_2|}{|e_1-{\left|v\right|}^{q-2}v|}},\hfill \end{array}$$

(11)

Then, Lemma 2 is proved.  □

3. The First Main Result

In this section, we present our first main result concerning the existence and the uniqueness of a solution. For simplicity, we denote the space ${\mathcal{H}}_{\mathcal{K},0}^{\alpha ,\nu ;\psi }\left(\mathrm{\Omega }\right)$ by X, and we assume that the function f is nontrivial in $X^{*}$.

Definition 1.

We say that $u\in X$ is a weak solution to problem (1) if for all $\phi \in X$, we have

$$\begin{array}{c}\hfill \mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),\phi \rangle ={\int }_{\mathrm{\Omega }}f\phi dx.\end{array}$$

(12)

Now, let the functional $\mathcal{G}:X\to \mathbb{R}$ be defined by

$$\mathcal{G}\left(u\right):=\widehat{\mathcal{M}}\left({\xi }_{\mathcal{K}}\left(u\right)\right)={\int }_0^{{\xi }_{\mathcal{K}}\left(u\right)}\mathcal{M}\left(s\right)ds,$$

and let the operator $\mathcal{H}:X\to X^{*}$ define by

$$\langle \mathcal{H}\left(u\right),\phi \rangle =\langle {\mathcal{G}}^{\prime }\left(u\right),\phi \rangle =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),\phi \rangle ,\mathrm{for}\mathrm{all}u,\phi \in X,$$

(13)

Remark 2.

By (12) and (13), $u\in X$ is a solution to problem (1) if and only if u is a solution to problem $\mathcal{H}u=f$.

The following result is crucial for the rest of this paper.

Proposition 9

([38,40,41]). Let E be a reflexive real Banach space. Let $\mathcal{K}:E\to E^{*}$ be an operator such that:

$\left(1\right)$

$\mathcal{K}$ is coercive.

$\left(2\right)$

$\mathcal{K}$ is hemicontinuous; that is, $\mathcal{K}$ is directionally weakly continuous, if the function

$$A\left(s\right)=\langle \mathcal{K}(u+sw),v\rangle$$

is continuous in s on $[0,1]$ for every $u,w,v\in X$.

$\left(3\right)$

For all $u,v\in X$, we have

$$\langle \mathcal{K}\left(u\right)-\mathcal{K}\left(v\right),u-v\rangle \ge 0.$$

(14)

Then, the equation

$$\mathcal{K}u=g$$

(15)

has at least one solution $u\in E$ for every $g\in E^{*}$. If, the inequality (14) is strict for all $u,v\in E$, with $u\ne v$, then the Equation (15) has a unique solution $u\in E$ for every $g\in E^{*}$.

Here is the initial main result.

Theorem 1.

Assume that the hypotheses $\left(H_1\right)–\left(H_3\right)$ are satisfied. Then, for any $f\in X^{*}$, the problem (1) has a unique solution $u\in X$.

To prove Theorem 1, we use the Proposition 9. For this, we need to prove several Lemmas.

Lemma 3.

Under hypotheses $\left(H_1\right)–\left(H_3\right)$, the operator $\mathcal{H}$ is coercive.

Proof. 

Using hypothesis $\left(H_3\right)$ and Lemma 1, we obtain

$$\langle \mathcal{H}\left(u\right),u\rangle =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right){\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\ge {\displaystyle \frac{m_0}{p^{-}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{p^{-}}.$$

(16)

from the fact that $p^{-}>2$, we deduce the result of Lemma 3.  □

Lemma 4.

Under hypotheses $\left(H_1\right)–\left(H_3\right)$, the operator $\mathcal{H}$ is hemicontinuous.

Proof. 

Let $t_1,t_2\in [0,1]$, and $u,v\in X$, then we have

$$\begin{array}{cc}\hfill & |\varphi \left(t_1\right)-\varphi \left(t_2\right)| = |\langle \mathcal{H}(u+t_{1}v)-\mathcal{H}(u+t_{2}v),\phi \rangle |\hfill \\ \hfill & \le {\int }_{\mathrm{\Omega }}|{\mathcal{Y}}_{t_1}{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)\hfill \\ \hfill & -{\mathcal{Y}}_{t_2}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v){|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & +{\int }_{\mathrm{\Omega }}\mu \left(x\right)|{\mathcal{Y}}_{t_1}{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)\hfill \\ \hfill & -{\mathcal{Y}}_{t_2}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v){|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx.\hfill \end{array}$$

(17)

where we note

$${\mathcal{Y}}_{t_1}=\mathcal{M}\left({\xi }_{\mathcal{K}}(u+t_{1}v)\right),\mathrm{and}{\mathcal{Y}}_{t_2}=\mathcal{M}\left({\xi }_{\mathcal{K}}(u+t_{2}v)\right).$$

From Lemma 2, we have

$$\begin{array}{cc}\hfill & |\varphi \left(t_1\right)-\varphi \left(t_2\right)|\hfill \\ \hfill & \le {\int }_{\mathrm{\Omega }}\left\{||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)-{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|\right.\hfill \\ \hfill & \left.\times (K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\right\}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx+K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{p\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & +{\int }_{\mathrm{\Omega }}\mu \left(x\right)\left\{||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)-{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|\right.\hfill \\ \hfill & \left.\times (K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\right\}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx+K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{q\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & \le (K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1}){\int }_{\mathrm{\Omega }}|{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)\hfill \\ \hfill & {-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v){|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1}){\int }_{\mathrm{\Omega }}\mu \left(x\right)|{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)\hfill \\ \hfill & {-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v){|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{p\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ & +K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{q\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx,\hfill \end{array}$$

(18)

where $K_{t_1}^{t_2}=|{\mathcal{Y}}_{t_1}-{\mathcal{Y}}_{t_2}|$.

Now, using the fact that for any $1<n<\infty$, there is a constant $c_n>0$ such that

$${\left(\right|x|}^{n-2}x-{\left|y\right|}^{n-2}y)\le c_n{|x-y|\left(\right|x|+|y\left|\right)}^{n-2},\forall x,y\in {\mathbb{R}}^N.$$

(19)

We deduce that

$$\begin{array}{cc}\hfill & |\varphi \left(t_1\right)-\varphi \left(t_2\right)|\hfill \\ \hfill & \le 2^{p^{+}-1}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\hfill \\ \hfill & \times {\int }_{\mathrm{\Omega }}\left(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{p\left(x\right)-2}+{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|}^{p\left(x\right)-2}\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & +2^{q^{+}-1}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\hfill \\ \hfill & \times {\int }_{\mathrm{\Omega }}\mu \left(x\right)\left(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{q\left(x\right)-2}+{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|}^{q\left(x\right)-2}\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & +K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{p\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ & +K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v){|}^{q\left(x\right)-1}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }\phi |dx,.\hfill \end{array}$$

(20)

On the other hand, since $t_1,t_2\in [0,1]$, then we obtain

$$|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{1}v)|\le |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|+|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|,$$

$$|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u+t_{2}v)|\le |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|+|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|,$$

and

$$|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|\le |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|+|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|.$$

Now, if we put $|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|+|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|=\xi$. Then, from the Hölder inequality, Propositions 2 and 7, we obtain

$$\begin{array}{cc}\hfill & |\varphi \left(t_1\right)-\varphi \left(t_2\right)|\hfill \\ \hfill & \le 2^{p^{+}}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1}){\int }_{\mathrm{\Omega }}{|\xi |}^{p\left(x\right)-2}|\xi ||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & +2^{q^{+}}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1}){\int }_{\mathrm{\Omega }}\mu \left(x\right){|\xi |}^{q\left(x\right)-2}|\xi |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx+K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}{|\xi |}^{p\left(x\right)-1}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ & +K_{t_1}^{t_2}{\int }_{\mathrm{\Omega }}{|\xi |}^{q\left(x\right)-1}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |dx\hfill \\ \hfill & \le 2^{q^{+}}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\left({|\xi |}_{p\left(x\right)}^{p^{+}-1}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{\phi |}_{p\left(x\right)}+{|\mu |}_{\infty }{|\xi |}_{q\left(x\right)}^{q^{+}-1}{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }\phi |}_{q\left(x\right)}\right)\hfill \\ \hfill +& K_{t_1}^{t_2}\left({\parallel \xi \parallel }_{1,\mathcal{K},0}^{p^{+}-1}+{\parallel \xi \parallel }_{1,\mathcal{K},0}^{q^{+}-1}\right){\parallel \phi \parallel }_{1,\mathcal{K},0}\hfill \\ \hfill & \le 2^{q^{+}+1}|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\left({\parallel \xi \parallel }_{1,\mathcal{K},0}^{p^{+}-1}+{|\mu |}_{\infty }{\parallel \xi \parallel }_{1,\mathcal{K},0}^{q^{+}-1}\right){\parallel \phi \parallel }_{1,\mathcal{K},0}\hfill \\ \hfill +& K_{t_1}^{t_2}\left({\parallel \xi \parallel }_{1,\mathcal{K},0}^{p^{+}-1}+{\parallel \xi \parallel }_{1,\mathcal{K},0}^{q^{+}-1}\right){\parallel \phi \parallel }_{1,\mathcal{K},0}.\hfill \end{array}$$

(21)

From hypothesis $\left(H_3\right)$ and Proposition 8, we get

$$K_{t_1}^{t_2}=|{\mathcal{Y}}_{t_1}-{\mathcal{Y}}_{t_2}|\to 0,\mathrm{and}{\mathcal{Y}}_{t_1}\to {\mathcal{Y}}_{t_2}\in [m_0,\infty )\mathrm{as}{t}_1\to t_2.$$

Then, $|t_1-t_2|(K_{t_1}^{t_2}+{\mathcal{Y}}_{t_1})\to 0$ as $t_1\to t_2$. Finally, we have

$$|\varphi \left(t_1\right)-\varphi \left(t_2\right)|=|\langle \mathcal{H}(u+t_{1}v)-\mathcal{H}(u+t_{2}v),\phi \rangle |\to 0\mathrm{as}{t}_1\to t_2.$$

So, we conclude that $\mathcal{H}$ is hemicontinuous.  □

Lemma 5.

Under hypotheses $\left(H_1\right)–\left(H_3\right)$, the operator $\mathcal{H}$ is strictly monotone.

Proof. 

Let $u,v\in X$ such that $u\ne v$ and ${\xi }_{\mathcal{K}}\left(u\right)\ge {\xi }_{\mathcal{K}}\left(v\right)$. So, from $\left(H_3\right)$ and Proposition 8, we get $\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\ge \mathcal{M}\left({\xi }_{\mathcal{K}}\left(v\right)\right)$.

• Firstly, using this inequality: ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\le 2^{-1}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2+|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)$, we deduce that

$$\begin{array}{cc}\hfill & \langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),u-v\rangle \hfill \\ \hfill & ={\int }_{\mathrm{\Omega }}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u+\mu \left(x\right){|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u)·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u-v)dx\hfill \\ \hfill & ={\int }_{\mathrm{\Omega }}\left\{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)}+\mu \left(x\right){|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)}\right.\hfill \\ \hfill & \left.-\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2}){}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\right\}dx\hfill \\ \hfill & \ge 2^{-1}{\int }_{\mathrm{\Omega }}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{\left|u\right|}^{q\left(x\right)-2}\left)\right(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx.\hfill \end{array}$$

(22)

Similarly, we obtain

$$\begin{array}{cc}\hfill & \langle {\xi }_{\mathcal{K}}^{\prime }\left(v\right),v-u\rangle \hfill \\ \hfill & ={\int }_{\mathrm{\Omega }}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v)·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(v-u)dx\hfill \\ \hfill & ={\int }_{\mathrm{\Omega }}\left\{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)}+\mu \left(x\right){\left|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\right|}^{q\left(x\right)}\right.\hfill \\ \hfill & \left.-\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2}){}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\right\}dx\hfill \\ \hfill & \ge 2^{-1}{\int }_{\mathrm{\Omega }}\left(\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{q\left(x\right)-2}\left)\right(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^2)dx,\hfill \end{array}$$

(23)

Now, let ${\mathrm{\Omega }}_{+}=\{x\in \mathrm{\Omega }:|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|\ge |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\left|\right\}$ and ${\mathrm{\Omega }}_{-}=\{x\in \mathrm{\Omega }:|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|<|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\left|\right\}$.

Next, by (22), (23) and $\left(H_3\right)$, we have

$$\begin{array}{cc}\hfill I_{+}\left(u\right):& =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),u-v\rangle \hfill \\ \hfill & =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right){\int }_{\mathrm{\Omega }}\left\{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)}+\mu \left(x\right){|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)}\right.\hfill \\ \hfill & \left.-(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2}){}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v\right\}dx\hfill \\ \hfill & \ge {\displaystyle \frac{\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)}{2}}{\int }_{{\mathrm{\Omega }}_{+}}(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{q\left(x\right)-2}\left)\right(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx\hfill \\ \hfill & -{\displaystyle \frac{\mathcal{M}\left({\xi }_{\mathcal{K}}\left(v\right)\right)}{2}}{\int }_{{\mathrm{\Omega }}_{+}}(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{q\left(x\right)-2}\left)\right(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx\hfill \\ \hfill & \ge {\displaystyle \frac{\mathcal{M}\left({\xi }_{\mathcal{K}}\left(v\right)\right)}{2}}{\int }_{{\mathrm{\Omega }}_{+}}\left\{(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2})\right.\hfill \\ \hfill & \left.-(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2})\right\}(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx\hfill \\ \hfill & \ge {\displaystyle \frac{m_0}{2}}{\int }_{{\mathrm{\Omega }}_{+}}\left\{(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2})\right.\hfill \\ \hfill & \left.-(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2})\right\}(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

(24)

and similarly, we have

$$\begin{array}{cc}\hfill I_{-}\left(v\right):& =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(v\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(v\right),v-u\rangle \hfill \\ \hfill & =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(v\right)\right){\int }_{\mathrm{\Omega }}\left\{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}^{p\left(x\right)}+\mu \left(x\right){|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|}^{q\left(x\right)}\right.\hfill \\ \hfill & \left.-(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2){}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u·{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v}\right\}dx\hfill \\ \hfill & \ge {\displaystyle \frac{m_0}{2}}{\int }_{{\mathrm{\Omega }}_{-}}\left\{(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)-2}+\mu {\left(x\right)}_1|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2})\right.\hfill \\ \hfill & \left.-(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{p\left(x\right)-2}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^{q\left(x\right)-2})\right\}(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^2-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v{|}^2)dx\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

(25)

Finally, we get

$$\begin{array}{c}\hfill \langle \mathcal{H}\left(u\right)-\mathcal{H}\left(v\right),u-v\rangle =I_{+}\left(u\right)+I_{-}\left(v\right)\ge 0.\end{array}$$

(26)

On the other hand, if $\langle \mathcal{H}\left(u\right)-\mathcal{H}\left(v\right),u-v\rangle =0$, we obtain $I_{+}\left(u\right)=I_{-}\left(v\right)=0$. So, we get ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{=}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v$, that is, ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }(u-v)=0$, which implies $\Vert u-{v\Vert }_{1,\mathcal{K},0}=0$, this contradicts the assumption that $u\ne v$ in X. So, we conclude that

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

(27)

  □

Proof of Theorem 1.

From Lemmas 3–5 and Proposition 9, the equation $\mathcal{H}u=f$ has a unique nontrivial solution $u_{*}\in X$, then problem (1) has a unique nontrivial solution $u_{*}\in X$.  □

4. The Second Main Result

In the next, we suppose that problem (1) has to do with the convection term, more precisely, the nonlinearity $f=f(x,u{,}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u)$, and we will prove that Equation (1) admits a weak solution in $X=X$.

Definition 2.

We say that $u\in X$ is a weak solution to problem (1) if for all $\phi \in X$ we have

$$\begin{array}{c}\hfill \mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),\phi \rangle ={\int }_{\mathrm{\Omega }}f(x,u{,}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u)\phi dx.\end{array}$$

(28)

Firstly, we define the operator $\mathcal{T}:X\to X^{*}$ by:

$$\langle \mathcal{T}\left(u\right),\phi \rangle :={\int }_{\mathrm{\Omega }}f(x,u{,}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u)\phi dx,$$

(29)

and the operator $\mathcal{H}$ by $\langle \mathcal{H}\left(u\right),\phi \rangle =\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)\langle {\xi }_{\mathcal{K}}^{\prime }\left(u\right),\phi \rangle =\langle {\mathcal{G}}^{\prime }\left(u\right),\phi \rangle$, where

$$\langle \mathcal{G}\left(u\right),\phi \rangle :=\widehat{\mathcal{M}}\left({\xi }_{\mathcal{K}}\left(u\right)\right)={\int }_0^{{\xi }_{\mathcal{K}}\left(u\right)}\mathcal{M}\left(s\right)ds.$$

Finally, we define the operator $\mathcal{A}:X\to X^{*}$ by

$$\langle \mathcal{A}\left(u\right),\phi \rangle :=\langle \mathcal{H}\left(u\right),\phi \rangle -\langle \mathcal{T}\left(u\right),\phi \rangle ,forallu,\phi \in X.$$

(30)

In the next, we will solve the operator equation

$$\mathcal{A}u=\mathcal{H}u-\mathcal{T}u=0.$$

(31)

In the following, we assume the following conditions for the nonlinearity f are verified:

$\left(S_1\right)$

$f:\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^3\to \mathbb{R}$ is a Carathéodory function such that $f(·,0,0)\ne 0$.

$\left(S_2\right)$

There exist $g\in L^{s^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)$ and $a_1,a_2>0$ satisfying

$$\left|f(x,y,z)\right|\le g\left(x\right)+a_1{\left|y\right|}^{s\left(x\right)-1}+a_2{\left|z\right|}^{p\left(x\right){\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}},$$

for all $(y,z)\in \mathbb{R}\times {\mathbb{R}}^N$, $s\in C_{+}\left(\overline{\mathrm{\Omega }}\right)$ and $s^{\prime }\left(x\right):={\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}$ and $s\left(x\right)<p_{\alpha }^{*}\left(x\right)$.

$\left(S_3\right)$

There exist $h\in L^{p^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)$, ${\beta }_1\in L^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-{\tau }^{-}}}}\left(\mathrm{\Omega }\right)$, and ${\beta }_2\in L^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-\tau \left(x\right)}}}\left(\mathrm{\Omega }\right)$ satisfying

$$\underset{\left|z\right|\to +\infty }{lim\; sup}{\displaystyle \frac{f(x,y,z)}{h\left(x\right)+{\beta }_1{\left(x\right)\left|y\right|}^{{\tau }^{-}-1}+{\beta }_2\left(x\right){\left|z\right|}^{\tau \left(x\right)-1}}}\le \lambda ,$$

for all $(y,z)\in \mathbb{R}\times {\mathbb{R}}^N$, $\tau \in C_{+}\left(\overline{\mathrm{\Omega }}\right)$, ${\tau }^{+}<p^{-}$, and $\lambda >0$.

$\left(S_4\right)$

$$m_0\le \mathcal{M}\left(t\right)\le \kappa (1+t^{\gamma -1}),$$

(32)

where $m_0,\kappa >0$ and $\gamma >1$.

Example 1.

Let $f:\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ be defined by:

$$f(x,y,z)=a\left(x\right)sin(y+1)+{b\left|y\right|}^{s\left(x\right)-1}{e}^{-\left|y\right|}+{c\left|z\right|}^{p\left(x\right)-1}ln(1+|z\left|\right),$$

where $a\in L^{s^{*}\left(x\right)}\left(\mathrm{\Omega }\right)$, $b,c>0$.

To show our result, we need the following proposition:

Proposition 10

([38,40,41]). Let E be a reflexive real Banach space and $\mathrm{\Psi }:E\to E^{*}$ be a pseudomonotone, bounded, and coercive operator, and $\varphi \in E^{*}$. Then, a solution of the equation $\mathrm{\Psi }u=\varphi$ exists.

Theorem 2.

Assume the assumptions $\left(S_1\right)$–$\left(S_4\right)$ are satisfied. Then, the problem (1) has at least one nontrivial solution $u\in X$.

Next, we show Theorem 2. For this, we prove the following lemma.

Lemma 6.

Under $\left(S_1\right)$–$\left(S_4\right)$, the operator $\mathcal{A}$ is coercive.

Proof. 

Without loss of generality, we assume that ${\Vert u\Vert }_{1,\mathcal{K},0}>1$. So, we obtain that ${\parallel u\parallel }_{1,\mathcal{K},0}=\parallel |{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|\parallel }_{\mathcal{K}}$. Hence, from the monotonicity of ${\rho }_{\mathcal{K}}$ and using Lemma 1, we obtain

$${\parallel u\parallel }_{1,\mathcal{K},0}\ge {\left({\rho }_{\mathcal{K}}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right)\right)}^{1/q^{+}}>1.$$

So, from $\left(S_3\right)$, and for ${\left|\right|u\left|\right|}_{1,\mathcal{K},0}$ big enough, we obtain

$$|f(x,u,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u)|\le \lambda \left(\left|h\right|+|{\beta }_1{\left|\right|u|}^{{\tau }^{-}-1}+|{\beta }_2\left|\right|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{\tau \left(x\right)-1}\right).$$

(33)

By Hölder inequality, Proposition 2, and embedding theorem, we have

$$\begin{array}{cc}\hfill \langle \mathcal{T}\left(u\right),u\rangle & \le {\int }_{\mathrm{\Omega }}\lambda \left(|h||u|+|{\beta }_1{||u|}^{{\tau }^{-}-1}|u|+|{\beta }_2||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{\tau \left(x\right)-1}|u|\right)dx\hfill \\ \hfill & \le \lambda \left({|h|}_{p^{\prime }}{|u|}_{p\left(x\right)}+|{\beta }_1{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-{\tau }^{-}}}}{||u|}^{{\tau }^{-}-1}{|}_{{\displaystyle \frac{q\left(x\right)}{{\tau }^{-}-1}}}{|u|}_{q\left(x\right)}\right.\hfill \\ \hfill & \left.+|{\beta }_2{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-\tau \left(x\right)}}}||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{\tau \left(x\right)-1}{|}_{{\displaystyle \frac{q\left(x\right)}{\tau \left(x\right)-1}}}{|u|}_{q\left(x\right)}\right)\hfill \\ \hfill & \le \lambda \left(c_1{\parallel u\parallel }_{1,\mathcal{K},0}^{{\tau }^{+}}+c_2{\parallel u\parallel }_{1,\mathcal{K},0}^{{\tau }^{-}}+c_3{\parallel u\parallel }_{1,\mathcal{K},0}\right),\hfill \end{array}$$

(34)

so

$${\displaystyle \frac{\langle \mathcal{T}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}\le \lambda \left(c_1{\parallel u\parallel }_{1,\mathcal{K},0}^{{\tau }^{+}-1}+c_2{\parallel u\parallel }_{1,\mathcal{K},0}^{{\tau }^{-}-1}+c_3\right).$$

(35)

By Lemma 3, we get

$${\displaystyle \frac{\langle \mathcal{H}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}\ge {\displaystyle \frac{m_0}{q^{+}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{p^{-}-1}.$$

(36)

Using (35) and (36), we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{\langle \mathcal{A}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}& \ge {\displaystyle \frac{\langle \mathcal{H}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}-{\displaystyle \frac{\langle \mathcal{T}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}\hfill \\ \hfill & \ge {\displaystyle \frac{m_0}{q^{+}}}{\parallel u\parallel }_{1,\mathcal{K},0}^{p^{-}-1}-\lambda \left(c_1{\parallel u\parallel }_{1,\mathcal{H},0}^{{\tau }^{+}-1}+c_2{\parallel u\parallel }_{1,\mathcal{H},0}^{{\tau }^{-}-1}+c_3\right),\hfill \end{array}$$

(37)

then

$$\begin{array}{c}\hfill \underset{{\parallel u\parallel }_{1,\mathcal{K},0}\to \infty }{lim}{\displaystyle \frac{\langle \mathcal{A}\left(u\right),u\rangle }{{\parallel u\parallel }_{1,\mathcal{K},0}}}=+\infty .\end{array}$$

(38)

  □

Lemma 7.

Under $\left(S_1\right)$–$\left(S_4\right)$, the operator $\mathcal{H}$ is continuous and bounded.

Proof. 

From $\left(S_4\right)$, $\mathcal{M}$ is continuous. On the other hand, from the fact that ${\xi }_{\mathcal{K}}\in C^1(X,X^{*})$, we deduce that $\mathcal{M}\left({\xi }_{\mathcal{K}}(·)\right)$ is continuous. Let $\left(u_n\right)\subset X$ be a sequence such that $u_n\to u\in X$. Then, from Lemma 2, we obtain

$$\begin{array}{cc}\hfill & |\langle \mathcal{H}\left(u_n\right)-\mathcal{H}\left(u\right),v\rangle |\hfill \\ \hfill & \le {\int }_{\mathrm{\Omega }}|{\mathcal{M}}_{u_n}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-{\mathcal{M}}_u|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +{\int }_{\mathrm{\Omega }}\mu \left(x\right)|{\mathcal{M}}_{u_n}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-{\mathcal{M}}_u{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & \le {\int }_{\mathrm{\Omega }}\left\{{||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|\times (K_{u_n}^u+{\mathcal{M}}_{u_n})+K_{u_n}^u\right\}\hfill \\ & {|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +{\int }_{\mathrm{\Omega }}\mu \left(x\right)\left\{{||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|\times (K_{u_n}^u+{\mathcal{M}}_{u_n})+K_{u_n}^u\right\}\hfill \\ & {|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & \le (K_{u_n}^u+{\mathcal{M}}_{u_n}){\int }_{\mathrm{\Omega }}{||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +(K_{u_n}^u+{\mathcal{M}}_{u_n}){\int }_{\mathrm{\Omega }}\mu \left(x\right)||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|dx\hfill \\ \hfill & +2K_{u_n}^u{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|dx,\hfill \end{array}$$

(39)

we denote ${\mathcal{M}}_{u_n}=\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u_n\right)\right)$, ${\mathcal{M}}_u=\mathcal{M}\left({\xi }_{\mathcal{K}}\left(u\right)\right)$, and $K_{u_n}^u=|{\mathcal{M}}_{u_n}-{\mathcal{M}}_u|$. On the other hand, from the embedding $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\hookrightarrow L_{\mu }^{q\left(x\right)}\left(\mathrm{\Omega }\right)$ and Propositions 5 and 7, we obtain

$$\begin{array}{cc}\hfill & |\langle \mathcal{H}\left(u_n\right)-\mathcal{H}\left(u\right),v\rangle |\hfill \\ \hfill & \le (K_{u_n}^u+{\mathcal{M}}_{u_n})|{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|{|}_{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|}_{p\left(x\right)}\hfill \\ \hfill & +(K_{u_n}^u+{\mathcal{M}}_{u_n})Z(u_n,u)|\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}{||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{v|}_{q\left(x\right)}+2K_{u_n}^u|\mathrm{\Omega }||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v|}_{p\left(x\right)},\hfill \end{array}$$

(40)

where

$$Z(u_n,u)=|\mu {\left(x\right)}^{{\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2} {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}.$$

Now, using the fact that $u_n\to u$ in X and from the embeddings

$$L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\hookrightarrow L_{\mu }^{q\left(x\right)}\left(\mathrm{\Omega }\right),X\hookrightarrow L^{\mathcal{K}}\left(\mathrm{\Omega }\right),\mathrm{and}X\hookrightarrow L^{p\left(x\right)}\left(\mathrm{\Omega }\right),$$

we obtain

$$\underset{n\to \infty }{lim}{\int }_{\mathrm{\Omega }}|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)}dx={\int }_{\mathrm{\Omega }}{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{p\left(x\right)}dx,$$

(41)

and

$$\underset{n\to \infty }{lim}{\int }_{\mathrm{\Omega }}{\mu \left(x\right)|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)}dx={\int }_{\mathrm{\Omega }}\mu \left(x\right){{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)}dx,$$

(42)

On the other hand, by Vitali’s Theorem (see [42] (Theorem 8)), we have $|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|\to |}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|$, $\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|\to \mu \left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u|$ in measure in $\mathrm{\Omega }$ and using the inequalities

$$\begin{array}{ccc}\hfill |{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2} {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n& -& {|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2} {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\hfill \\ & \le & 2^{{\displaystyle \frac{p^{+}}{p^{-}-1}}-1}{(|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)}{+|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)}),\hfill \end{array}$$

(43)

and

$$\begin{array}{ccc}\hfill {\mu \left(x\right)||}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2} {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n& -& {|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\hfill \\ & \le & 2^{{\displaystyle \frac{q^{+}}{q^{-}-1}}-1}{\mu \left(x\right)(|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)}{+|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)}).\hfill \end{array}$$

(44)

We get that the families

$$\left\{||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\right\},$$

(45)

and

$$\left\{\mu \left(x\right)||{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\right\},$$

(46)

are uniformly integrable in $\mathrm{\Omega }$. So, from Vitali’s Theorem, we conclude that the functions

$$|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u,\mathrm{and}\mu \left(x\right){|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u,$$

are integrable. Moreover

$$|\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{-|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u{|}^{p\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|{|}_{{\displaystyle \frac{p\left(x\right)}{p\left(x\right)-1}}}\to 0,$$

and

$$|\mu {\left(x\right)}^{{\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}}\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n-|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)-2}{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\to 0.$$

Now, by combining condition $\left(f_4\right)$ with Proposition 8, we obtain

$$K_{u_n}^u\to 0,\mathrm{and}{\mathcal{M}}_{u_n}\to {\mathcal{M}}_u\in [m_0,\infty ).$$

So, we conclude that

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

(47)

This implies that $\mathcal{H}\in C(X,\mathbb{R})$. Next, we show that $\mathcal{H}$ is bounded. For this, Let $u,v\in X\setminus \left\{0\right\}$, then from $\left(S_4\right)$ and Lemma 1, we obtain

$$\begin{array}{cc}\hfill & min\left\{{\displaystyle \frac{1}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}^{p^{-}-1}}},{\displaystyle \frac{1}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}^{q^{+}-1}}}\right\}〈\mathcal{H}\left(u\right),{\displaystyle \frac{v}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}〉\hfill \\ \hfill & \le \kappa {\rho }_{\mathcal{K}}^{\gamma -1}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right){\int }_{\mathrm{\Omega }}\left(|{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}}}{|}^{p\left(x\right)-1}{\displaystyle \frac{{|}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }v|}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}\right.\hfill \\ \hfill & +\mu {\left(x\right)}^{{\displaystyle \frac{q\left(x\right)-1}{q\left(x\right)}}}\left|{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}}}{|}^{q\left(x\right)-1}\mu {\left(x\right)}^{{\displaystyle \frac{1}{q\left(x\right)}}}{\displaystyle \frac{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v|}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}\right)dx\hfill \\ \hfill & \le \kappa {\rho }_{\mathcal{K}}^{\gamma -1}\left({}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\right){\int }_{\mathrm{\Omega }}\left({\displaystyle \frac{p^{+}-1}{p^{-}}}|{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}}}{|}^{p\left(x\right)}+{\displaystyle \frac{1}{p^{-}}}{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v}{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}{|}^{p\left(x\right)}\right.\hfill \\ \hfill & +{\displaystyle \frac{\mu \left(x\right)(q^{+}-1)}{q^{-}}}\left|{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}}}{|}^{q\left(x\right)}{\displaystyle \frac{\mu \left(x\right)}{q^{-}}}\right|{\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}{|}^{q\left(x\right)})dx\hfill \\ \hfill & \le \kappa {\rho }_{\mathcal{K}}^{\gamma -1}{(}^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u)\left({\displaystyle \frac{q^{+}-1}{p^{-}}}{\rho }_{\mathcal{K}}\left({\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}}}\right)+{\displaystyle \frac{1}{p^{-}}}{\rho }_{\mathcal{K}}\left({\displaystyle \frac{{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }v}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{\kappa q^{+}}{p^{-}}}{{\parallel }^{\mathrm{H}}{\mathbb{D}}_{0+}^{\alpha ,\nu ;\psi }u\parallel }_{\mathcal{K}}^{(\gamma -1)q^{+}}.\hfill \end{array}$$

(48)

So, we get

$$\begin{array}{cc}\hfill {\parallel \mathcal{H}\left(u\right)\parallel }_{X^{*}}& =\underset{v\in X\setminus \left\{0\right\}}{sup}{\displaystyle \frac{\langle \mathcal{H}\left(u\right),v\rangle }{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{v\parallel }_{\mathcal{K}}}}\le {\displaystyle \frac{\kappa q^{+}}{p^{-}}}\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}^{(\gamma -1)q^{+}}max\{\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u\parallel }_{\mathcal{K}}^{p^{-}-1},\parallel {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{\parallel }_{\mathcal{K}}^{q^{+}-1}\}.\hfill \end{array}$$

(49)

This implies that $\mathcal{H}$ is bounded.  □

Lemma 8.

Under $\left(S_1\right)$–$\left(S_4\right)$, the operator $\mathcal{T}$ is continuous and bounded.

Proof. 

Let $\mathrm{\Gamma }:X\to L^{s^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)$ be defined by

$$\mathrm{\Gamma }\left(u\right)=f(x,u,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u).$$

(50)

Let $u\in X$ such that ${\parallel u\parallel }_{1,\mathcal{K},0}\le 1$, using assumption $\left(S_4\right)$, the embeddings $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\hookrightarrow L^{s\left(x\right)}\left(\mathrm{\Omega }\right)$ and $X\hookrightarrow L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$, we get

$$\begin{array}{cc}\hfill & {\int }_{\mathrm{\Omega }}{|\mathrm{\Gamma }\left(u\right)|}^{s^{\prime }\left(x\right)}dx\hfill \\ & ={\int }_{\mathrm{\Omega }}{|f(x,u,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u)|}^{s^{\prime }\left(x\right)}dx\hfill \\ \hfill & \le {\int }_{\mathrm{\Omega }}|g\left(x\right)+a_1{|u|}^{s\left(x\right)-1}+a_2|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{{|}^{p\left(x\right){\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}|}^{s^{\prime }\left(x\right)}dx\hfill \\ \hfill & \le c{\int }_{\mathrm{\Omega }}\left({|g\left(x\right)|}^{s^{\prime }\left(x\right)}+{|u|}^{s\left(x\right)}+{|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u|}^{p\left(x\right)}\right)dx\hfill \\ \hfill & \le c{\int }_{\mathrm{\Omega }}\left({|g\left(x\right)|}^{s^{\prime }\left(x\right)}+{|u|}^{s\left(x\right)}+(|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u|}^{p\left(x\right)}+\mu \left(x\right)|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u{|}^{q\left(x\right)})\right)dx\hfill \\ \hfill & \le {c\parallel u\parallel }_{1,\mathcal{K},0}.\hfill \end{array}$$

(51)

Then, $\mathrm{\Gamma }$ is bounded in $L^{s^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)$. Next, let $u_n\to u$ in X. Then, we get ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n\to {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u$ in $L^{s\left(x\right)}{\left(\mathrm{\Omega }\right)}^N$. Then, there exists a subsequence, denoted by $\left(u_n\right)$, and two functions $V_1\left(x\right)\in L^{s\left(x\right)}\left(\mathrm{\Omega }\right)$ and $V_2\left(x\right)\in L^{s\left(x\right)}{\left(\mathrm{\Omega }\right)}^N$, such that:

• $u_n\left(x\right)\to u\left(x\right)$ and ${}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n\left(x\right)\to {}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\left(x\right)$ a. e in $\mathrm{\Omega }$,
• $|u_n\left(x\right)|\le V_1\left(x\right)$ and $|{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n\left(x\right)|\le |V_2\left(x\right)|$ a. e in $\mathrm{\Omega }$ for all n.

• From $\left(S_1\right)$, the function f is continuous in the second and third arguments, then we get

$$f(x,u_n\left(x\right),{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n\left(x\right))\to f(x,u\left(x\right),{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u\left(x\right))\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega }\mathrm{as}n\to \infty .$$

(52)

On the other hand, using $\left(S_2\right)$, we deduce that

$$\begin{array}{cc}\hfill |f(x,u_n\left(x\right),{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n\left(x\right))|& \le g\left(x\right)+a_1|V_1{\left(x\right)|}^{s\left(x\right)-1}+a_2{|V_2\left(x\right)|}^{p\left(x\right){\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}.\hfill \end{array}$$

(53)

From Proposition 2, we obtain

$${\int }_{\mathrm{\Omega }}{a}_1|V_1{|}^{s\left(x\right)-1}dx\le c||V_1{|}^{s\left(x\right)-1}{|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{|\mathbf{1}|}_{s\left(x\right)}\le c{|V_1|}_{s\left(x\right)}^{s^{+}-1},$$

(54)

and

$${\int }_{\mathrm{\Omega }}{a}_2|V_2{|}^{p\left(x\right){\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}dx\le c||V_2{|}^{p\left(x\right){\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}{|}_{{\displaystyle \frac{s\left(x\right)}{s\left(x\right)-1}}}{|\mathbf{1}|}_{s\left(x\right)}\le c{|V_2|}_{p\left(x\right)}^{p^{+}}.$$

(55)

On the other hand, using the embeddings $L^{\mathcal{K}}\left(\mathrm{\Omega }\right)\hookrightarrow L^{s\left(x\right)}\left(\mathrm{\Omega }\right)$, $X\hookrightarrow W_0^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)$, and $X\hookrightarrow L^{\mathcal{K}}\left(\mathrm{\Omega }\right)$, we concluded that the second term of (53) is integrable; then, by the Lebesgue dominated convergence theorem (see, [42]), we obtain

$$f(x,u_n,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n)\to f(x,u,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }u)\mathrm{in}{L}^1\left(\mathrm{\Omega }\right).$$

(56)

So, we get

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

(57)

Finally, by Proposition 1, we concluded that $\mathrm{\Gamma }$ is continuous in $L^{s^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)$. On the other hand, $i^{*}:L^{s^{\prime }\left(x\right)}\left(\mathrm{\Omega }\right)\to X^{*}$ is continuous, then $\mathcal{T}=i^{*}\circ \mathrm{\Gamma }$ is continuous and bounded.  □

Definition 3.

Let a bounded operator $\mathcal{A}:X\to X^{*}$; then, we say that $\mathcal{A}$ is a pseudomonotone operator if $v_n\rightharpoonup v$ in X and ${lim\; sup}_{n\to \infty }\langle \mathcal{A}\left(v_n\right),v_n-v\rangle \le 0$, then, $\mathcal{A}{v}_n\rightharpoonup \mathcal{A}v$ and $\langle \mathcal{A}\left(v_n\right),v_n\rangle \to \langle \mathcal{A}v,v\rangle$.

Lemma 9.

Under $\left(S_1\right)$–$\left(S_4\right)$, the operator $\mathcal{A}$ is pseudomonotone.

Proof. 

Let $\left(u_n\right)\subset X$ with

$$u_n\rightharpoonup u\in X\mathrm{and}\underset{n\to \infty }{lim\; sup}\langle \mathcal{A}\left(u_n\right),u_n-u\rangle \le 0.$$

(58)

Since $u_n\rightharpoonup u\in X$, then $\left(u_n\right)$ is ${\parallel ·\parallel }_{1,\mathcal{K},0}$-bounded. Since $\mathrm{\Gamma }$ is bounded and by the compact embedding $X\hookrightarrow L^{s\left(x\right)}\left(\mathrm{\Omega }\right)$, we get

$$\begin{array}{cc}\hfill \left|{\int }_{\mathrm{\Omega }}f(x,u_n,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n)(u_n-u)dx\right|& \le c|\mathrm{\Gamma }\left(u_n\right){|}_{{\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}{|u_n-u|}_{s\left(x\right)}\hfill \\ \hfill & \le c\underset{n\in \mathbb{N}}{sup}|\mathrm{\Gamma }\left(u_n\right){|}_{{\displaystyle \frac{s\left(x\right)-1}{s\left(x\right)}}}{|u_n-u|}_{s\left(x\right)}\to 0\mathrm{as}n\to \infty ,\hfill \end{array}$$

(59)

then

$$\underset{n\to \infty }{lim}{\int }_{\mathrm{\Omega }}f(x,u_n,{}^{\mathrm{H}}\mathbb{D}_{0+}^{\alpha ,\nu ;\psi }{u}_n)(u_n-u)dx=0.$$

(60)

On the other hand, using assumption $\left(S_4\right)$ and equality (30), we conclude that

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

(61)

Then, by Proposition 8, the operator $\mathcal{H}$ satisfies the $\left(S_{+}\right)$-property. So, by (58) and (61), we get $u_n\to u$ in X. Since $\mathcal{A}$ is continuous and bounded (Lemmas 7 and 8), we deduce that $\mathcal{A}\left(u_n\right)\to \mathcal{A}\left(u\right)$, finally, we deduce that $\mathcal{A}$ is pseudomonotone.  □

Proof of Theorem 2.

By combining Lemmas 6–8 with Lemma 9, we deduce that $\mathcal{A}$ is a pseudomonotone, bounded, and coercive operator. Hence, from Proposition 10, we conclude that the equation $\mathcal{A}u=\mathcal{H}u-\mathcal{T}u=0$ admits a solution $u_{*}$. Moreover, from assumptions $\left(S_1\right)$ and $\left(S_4\right)$, we infer that $u_{*}$ is nontrivial, and the proof is completed.  □

5. Conclusions

In this paper, I have succeeded in studying Kirchhoff’s two-phase problems involving the $\psi$-Hilfer fractional derivative and variable-exponent growth conditions. The investigation addressed two distinct mathematical challenges:

• First, I analyzed a class of problems with general nonlinear sources. By demonstrating that the associated functional energy is coercive, hemicontinuous, and strictly monotone, I proved the existence and the uniqueness of weak solutions through the application of monotone operator theory.
• Second, I extended the analysis to include convection-type nonlinearities, which inherently break the variational structure of the problem. In this context, I employed the theory of pseudomonotone operators to establish the existence of at least one weak solution under suitable growth and mixed-order assumptions.

The results of this study contribute to the mathematical modeling of complex physical phenomena, including nonlocal elasticity, viscoelasticity, and phase-transition media, where the $\psi$-Hilfer derivative provides a good tool for capturing tunable memory and hereditary effects.

Building upon the results presented here, several interesting directions for future research remain open:

• While this work focused on existence and uniqueness, exploring the multiplicity of solutions for these generalized Kirchhoff double-phase models using critical point theory or the LusternikSchnirelmann category remains a promising path.
• Developing efficient numerical schemes to approximate solutions for $\psi$-Hilfer fractional double-phase problems would bridge the gap between theoretical existence results and practical engineering applications.
• Investigating the global regularity and boundedness of weak solutions in Musielak–Orlicz–Sobolev spaces would provide deeper insights into the behavior of these nonlinear fractional systems.