Study of Generalized Double-Phase Problem with ς-Laplacian Operator

Abstract

In this paper, we explore a novel class of double-phase $\varsigma$-Laplacian problems involving a $\varphi$-Hilfer fractional operator. Employing variational techniques and weighted Musielak space theory, we establish the existence of infinitely many positive solutions under suitable assumptions on the nonlinearities. Our main results are original and significantly advance the literature on problems featuring $\varphi$-Hilfer derivatives and the $\varsigma$-Laplacian operator.

1. Introduction and Motivation

Fractional differentiation was considered closely aligned with traditional differentiation. However, modern research now focuses on viewing fractional differentiation as a more expansive concept than ordinary differentiation. In mathematical analysis, fractional analysis investigates different methods of defining the real or complex powers of the differentiation and integration operators. Fractional differential equations, which extend the idea of non-integer and generalized differential equations, appear in both time and space domains and include a power-law memory kernel that reflects nonlocal interactions [1].

There are several methods for introducing fractional integro-differential operators, including the Riemann–Liouville, the Caputo, the Hadamard and the Grunwald–Letnikov operators. It should be noted that most of the literature on fractional differentiation focuses primarily on the Riemann–Liouville and Caputo fractional derivatives. In addition, there are other well-established definitions, such as the Hadamard fractional derivative and the Erdélyi–Kober fractional derivative, among others. For more information on fractional calculus, interested readers are referred to [2,3].

In the context of modern physics and mechanics, it is crucial to consider the evolving landscape when developing mathematical models. Drawing from relevant studies, it is necessary to concentrate on specific areas to deepen our understanding of the theory underlying the central issue in this research. To this end, we employ the generalized $\varphi$-Hilfer fractional derivative to analyze a nonlinear Kirchhoff equation with a positive parameter. This equation is subject to Dirichlet boundary conditions and is given by the following:

$$\left\{\begin{array}{cc}{\mathbb{D}}_T^{\gamma ,\beta ;\varphi }\left({\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma -2} {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi +\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha -2} {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right)=g(x,\varpi ),\hfill & \mathrm{in} \mathfrak{Q},\hfill \\ \hfill \\ \varpi =0,\hfill & \mathrm{on} \partial \mathfrak{Q},\hfill \end{array}\right.$$

(1)

where ${\mathbb{D}}_T^{\gamma ,\beta ;\varphi }$ and ${\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }$ are $\varphi$-Hilfer fractional derivatives of order ${\displaystyle \frac{1}{\varsigma }}<\gamma <1$ and type $0\le \beta \le 1$, $\mathbf{b}(·)$ is a nonnegative weight function, and $g:\mathfrak{Q}\times \mathbb{R}⟶\mathbb{R}$ is a function of Carathéodory type such that for some $t_0>0$, we have

$${\displaystyle \underset{t\in \left[0,t_0\right]}{sup}g(·,t)\in L^{\infty }\left(\mathfrak{Q}\right).}$$

Moreover, we assume that g satisfies the following hypotheses:

H1.

There are two sequences ${\left\{u_k\right\}}_{k\in \mathbb{N}},{\left\{v_k\right\}}_{k\in \mathbb{N}}$, such that for any $k\in N$ we have

$${\displaystyle 0<u_k<v_k,\underset{k\to +\infty }{lim}{v}_k=0,}$$

and

$${\int }_0^{u_k}g(x,s)ds=\underset{t\in \left[u_k,v_k\right]}{sup}{\int }_0^{t}g(x,s)ds, for almost all x\in \mathfrak{Q}.$$

H2.

There exists a sequence ${\left\{w_k\right\}}_{k\in \mathbb{N}}\subset \left(0,v_k\right]$ such that

$$\underset{x\in \mathfrak{Q}}{essinf}{\int }_0^{w_k}g(x,s)ds>0.$$

We note the following function:

$$g(x,\varpi )=\left\{\begin{array}{cc}-\left(1+{\left|x\right|}^{\varsigma }\right)\left[\varsigma \varpi cos\left({\varpi }^{-\varsigma }\right)-(\varsigma +2){\varpi }^{\varsigma +1}sin\left({\varpi }^{-\varsigma }\right)\right],\hfill & \mathrm{if} \varpi >0,\hfill \\ 0,\hfill & \mathrm{otherwise}, \hfill \end{array}\right.$$

satisfies hypotheses $\left(\mathbf{H}\mathbf{1}\right)$ and $\left(\mathbf{H}\mathbf{2}\right)$. Indeed, a simple calculation implies that

$$G(x,\varpi )=\left\{\begin{array}{cc}\left(1+{\left|x\right|}^{\varsigma }\right){\varpi }^{\varsigma +2}sin\left({\varpi }^{-\varsigma }\right),\hfill & \mathrm{if} \varpi >0\hfill \\ 0,\hfill & \mathrm{otherwise}. \hfill \end{array}\right.$$

Let us define the sequences ${\left\{u_k\right\}}_{k\in \mathbb{N}},{\left\{v_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{z_k\right\}}_{k\in \mathbb{N}}$ by

$$u_k={\left(2\pi (k+1)\right)}^{{\displaystyle \frac{-1}{\varsigma }}}, v_k={\left((k+3/4)\pi \right)}^{{\displaystyle \frac{-1}{\varsigma }}}, z_k={\left((2k+1/4)\pi \right)}^{{\displaystyle \frac{-1}{\varsigma }}}.$$

It is not difficult to see that

$${\int }_0^{u_k}g(x,s)ds=\underset{t\in \left[u_k,v_k\right]}{sup}{\int }_0^{t}g(x,s)ds,$$

and

$$G\left(x,z_k\right)\ge 0.$$

This means that conditions $\left(\mathbf{H}\mathbf{1}\right)$ and $\left(\mathbf{H}\mathbf{2}\right)$ are fulfilled.

Related to double-phase problems, the stationary general reaction–diffusion double phase is given by the form

$$u_t=\mathrm{d}\mathrm{i}\mathrm{v}[A\left(u\right)\nabla u]+b(x,u), \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} A\left(u\right)={|\nabla u|}^{\varsigma -2}+{|\nabla u|}^{\alpha -2},$$

(2)

where the function u represents a concentration, and $\mathrm{d}\mathrm{i}\mathrm{v}\left[A\right(u)\nabla u]$ relates to diffusion with diffusion coefficient $A\left(u\right)$. The term $b(x,u)$ corresponds to sources and loss processes, and this type of problem has applications in physics and allied fields such as biophysics, plasma physics, solid-state physics, and chemical reaction design. For more information, refer to [4,5].

Recently, great attention has been devoted to the study of the following functional:

$$u\mapsto {\int }_{\mathfrak{Q}}\left({|\nabla u|}^{\varsigma }+a\left(x\right){|\nabla u|}^{\alpha }\right)dx,$$

(3)

with $1<\varsigma <\alpha$, $a(·)\ge 0$ which is related to the non-Newtonian fluid that can be characterized by Newton’s law of viscosity as the form,

$$F=\mu A{\displaystyle \frac{\partial u}{\partial y}},$$

where F is internal friction with the opposite direction of u, $\mu$ is the viscosity coefficient, and A is the contact area between the plate and fluid. More generally, we have

$$\overrightarrow{F}=\mu A\nabla u$$

where $\nabla u$ is the shear rate. By taking $\mu ={|\nabla u|}^{\varsigma -2}+a\left(x\right){|\nabla u|}^{\varsigma -2}$, we obtain the following non-Newtonian fluid equation:

$$\begin{array}{c}\hfill -\mathrm{d}\mathrm{i}\mathrm{v}\left(\left({|\nabla u|}^{\varsigma -2}+a\left(x\right){|\nabla u|}^{\varsigma -2}\right)\nabla u\right)=\lambda f,\end{array}$$

where $\lambda =1/A$ and $f=-\mathrm{d}\mathrm{i}\mathrm{v}\overrightarrow{F}$. For more details, we refer to [6]. Another example of this type of problem is given by Crespo et al. in [7] where they studied a double-phase problem characterized by the following operator:

$$\begin{array}{c}\hfill \mathrm{d}\mathrm{i}\mathrm{v}\left({|\nabla u|}^{\varsigma \left(x\right)}+a\left(x\right){|\nabla u|}^{\alpha \left(x\right)}\right),\end{array}$$

(4)

with $\varsigma ,\alpha \in C\left(\mathfrak{Q}\right)$ such that $1<\varsigma \left(x\right)<N$, $\varsigma \left(x\right)<\alpha \left(x\right)$ for all $x\in \mathfrak{Q}$ and $0\le \mu (·)\in L^1\left(\mathfrak{Q}\right)$. They proved certain properties of the operator (4) such as the continuity, strict monotonicity, $\left(S_{+}\right)$-property and showed the existence and uniqueness of corresponding elliptic equations. In [8], the authors studied the existence of two weak solutions for the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{d}\mathrm{i}\mathrm{v}\left({|\nabla u|}^{\varsigma \left(x\right)}+a\left(x\right){|\nabla u|}^{\alpha \left(x\right)}\right)={\lambda h\left(x\right)\left|u\right|}^{-\varsigma \left(x\right)}+\xi \left(x\right){\left|u\right|}^{s\left(x\right)-2}u\hfill & \mathrm{in} \mathfrak{Q},\hfill \\ \hfill \\ u=0\hfill & \mathrm{on} \partial \mathfrak{Q},\hfill \end{array}\right.\end{array}$$

where $\mathfrak{Q}\subset$${\mathit{R}}^N$, $N\ge 2$, is a bounded domain with smooth boundary $\partial \mathfrak{Q}$ and $\lambda >0$ is a real parameter. The functions $h\left(x\right),\xi \left(x\right)\in C\left(\overline{\mathfrak{Q}}\right)$ are positive with compact support in $\mathfrak{Q}$. They used the Nehari manifold method based on fibering maps to establish the existence results under suitable conditions on the functions $\varsigma$, $\alpha$, $\varsigma$, and s. For works involving this type of functional, we refer to the results discussed by Zhikov [9].

In [10], the authors established the existence of solutions to the following new class of singular double-phase $\varsigma$-Laplacian equation with a $\varphi$-Hilfer fractional operator combined from a parametric term, namely the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{c} {}^{\mathrm{H}}\mathrm{D}_T^{\alpha ,\beta ;\varphi }\left({\left|{}^{\mathrm{H}}\mathrm{D}_{0+}^{\alpha ,\beta ;\varphi }\mathbf{x}\right|}^{\varsigma -2}{}^{\mathrm{H}}\mathrm{D}_{0+}^{\alpha ,\beta ;\varphi }\mathbf{x}+\mu \left(t\right){\left|{}^{\mathrm{H}}\mathrm{D}_{0+}^{\alpha ,\beta ;\varphi }\mathbf{x}\right|}^{\alpha -2}{}^{\mathrm{H}}\mathrm{D}_{0+}^{\alpha ,\beta ;\varphi }\mathbf{x}\right)\hfill \\ =\xi \left(t\right){\mathbf{x}}^{-\sigma }+\lambda {\mathbf{x}}^{r-1}, \hspace{1em}\hspace{1em}\mathrm{in} \mathfrak{Q}=[0,T]\times [0,T],\hfill \\ \\ \hspace{1em}\hspace{1em}\hspace{1em}\mathbf{x}=0,\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{on} \partial \mathfrak{Q}.\hfill \end{array}\right.\end{array}$$

Our approach to proving the existence and multiplicity results for problem (1) relies on the utilization of the variational approach in appropriate $\varphi$-Hilfer fractional derivative spaces. For more details on these spaces, we refer to [11,12,13,14,15,16].

This work is organized as follows. In Section 2, we provide a brief overview of the key features of (weighted) Musielak spaces and $\varphi$-Hilfer fractional derivative spaces. Moving to Section 3, we present the existing solutions to problem (1), along with their corresponding proofs. Section 4 is reserved for a conclusion.

2. Preliminary Overview

In this section, we present a concise overview of the essential characteristics of (weighted) Musielak spaces and $\varphi$-Hilfer fractional derivative spaces. For more details, we refer to [15,17,18] Consider the nonlinear function $\mathcal{H}:\mathfrak{Q}\times {\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$ defined by

$$\mathcal{H}(x,\omega )={\omega }^{\varsigma }+\mathbf{b}\left(x\right){\omega }^{\alpha }.$$

Let $\mathbf{M}\left(\mathfrak{Q}\right)$ be the space of all measurable functions $\omega :\mathfrak{Q}⟶\mathbb{R}$. Then, Musielak space $L^{\mathcal{H}}\left(\mathfrak{Q}\right)$ is given by

$$L^{\mathcal{H}}\left(\mathfrak{Q}\right)=\left\{\omega \in \mathbf{M}\left(\mathfrak{Q}\right):{\varrho }^{\mathcal{H}}\left(\omega \right):={\int }_{\mathfrak{Q}}\mathcal{H}(x,|\omega \left|\right)\mathrm{d}x<\infty \right\},$$

equipped with the Luxemburg norm

$${\parallel \omega \parallel }_{L^{\mathcal{H}}\left(\mathfrak{Q}\right)}=inf\left\{\delta >0:{\varrho }^{\mathcal{H}}\left({\displaystyle \frac{\omega }{\delta }}\right)\le 1\right\}.$$

Moreover, we define the weighted space

$$L_{\mathbf{b}}^{\alpha }\left(\mathfrak{Q}\right)=\left\{\mathbf{f}\in \mathbf{M}\left(\mathfrak{Q}\right): {\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|\mathbf{f}\right|}^{\alpha }\mathrm{d}x<+\infty \right\},$$

with the semi norm

$${\parallel \mathbf{f}\parallel }_{\mathbf{b},\alpha }:={\left({\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|\mathbf{f}\right|}^{\alpha }\right)}^{{\displaystyle \frac{1}{q}}}.$$

Next, we present some results on the $\varphi$-Hilfer fractional derivative space. For this, let $A:=[c,d]$$(-\infty \le c<d\le \infty )$, $n-1<\gamma <n$, $n\in \mathbb{N}$, $\mathbf{f}$, $\varphi \in C^n(A,\mathbb{R})$ such that $\varphi$ is increasing and ${\varphi }^{\prime }\left(x\right)\ne 0$, for all $x\in A$.

• The left-sided fractional $\varphi$-Hilfer integrals of a function $\mathbf{f}$ are given by

$${\mathrm{I}}_{c^{+}}^{\gamma ;\varphi }\mathbf{f}\left(x\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_c^x{\varphi }^{\prime }\left(y\right){(\varphi \left(x\right)-\varphi \left(y\right))}^{\gamma -1}\mathbf{f}\left(y\right)\mathrm{d}y,$$

(5)

• The right-sided fractional $\varphi$-Hilfer integrals of a function $\mathbf{f}$ are given by

$${\mathrm{I}}_{d^{-}}^{\gamma ;\varphi }\mathbf{f}\left(x\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_x^d{\varphi }^{\prime }\left(y\right){(\varphi \left(y\right)-\varphi \left(x\right))}^{\gamma -1}\mathbf{f}\left(y\right)\mathrm{d}y.$$

(6)

• The left-sided $\varphi$-Hilfer fractional derivative for a function $\mathbf{f}$ of order $\gamma$ and type $0\le \beta \le 1$ is defined by

  $${\mathbb{D}}_{c^{+}}^{\gamma ,\beta ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{c^{+}}^{\beta (n-\gamma );\varphi }{\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(x\right)}}{\displaystyle \frac{d}{\mathrm{d}x}}\right)}^n{\mathrm{I}}_{c^{+}}^{(1-\beta )(n-\gamma );\varphi }\mathbf{f}\left(x\right),$$

• The right-sided $\varphi$-Hilfer fractional derivative for a function $\mathbf{f}$ of order $\gamma$ and type $0\le \beta \le 1$ is defined by

  $${\mathbb{D}}_{d^{-}}^{\gamma ,\beta ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{d^{-}}^{\beta (n-\gamma );\varphi }{\left(-{\displaystyle \frac{1}{{\varphi }^{\prime }\left(x\right)}}{\displaystyle \frac{d}{\mathrm{d}x}}\right)}^n{\mathrm{I}}_{d^{-}}^{(1-\beta )(n-\gamma );\varphi }\mathbf{f}\left(x\right).$$

Choosing $\beta \to 1$, we obtain $\varphi$-Caputo fractional derivatives left-sided and right-sided, given by

$${\mathbb{D}}_{c^{+}}^{\gamma ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{c^{+}}^{(n-\gamma );\varphi }{\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(x\right)}}{\displaystyle \frac{d}{\mathrm{d}x}}\right)}^n\mathbf{f}\left(x\right),$$

(7)

$${\mathbb{D}}_{d^{-}}^{\gamma ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{d^{-}}^{(n-\gamma );\varphi }{\left(-{\displaystyle \frac{1}{{\varphi }^{\prime }\left(x\right)}}{\displaystyle \frac{d}{\mathrm{d}x}}\right)}^n\mathbf{f}\left(x\right).$$

(8)

Remark 1.

The ϕ-Hilfer fractional derivatives defined as above can be written in the following form:

$${\mathbb{D}}_{c^{+}}^{\gamma ,\beta ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{c^{+}}^{\mu -\gamma ;\varphi }{\mathbb{D}}_{c^{+}}^{\gamma ;\varphi }\mathbf{f}\left(x\right),$$

and

$${\mathbb{D}}_{d^{-}}^{\gamma ,\beta ;\varphi }\mathbf{f}\left(x\right)={\mathrm{I}}_{d^{-}}^{\mu -\gamma ;\varphi }{\mathbb{D}}_{d^{-}}^{\gamma ;\varphi }\mathbf{f}\left(x\right),$$

with $\mu =\gamma +\beta (n-\gamma )$ and ${\mathrm{I}}_{c^{+}}^{\mu -\gamma ;\varphi }$, ${\mathrm{I}}_{d^{-}}^{\mu -\gamma ;\varphi }$, ${\mathbb{D}}_{c^{+}}^{\gamma ;\varphi }$ and ${\mathbb{D}}_{d^{-}}^{\gamma ;\varphi }$ as defined in (5)–(8).

Now that we have all the necessary tools, we are ready to commence our study. To facilitate this, we define the working space ${\mathbb{H}}_{\varsigma \left(x\right)}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right)$ as follows:

$${\mathbb{H}}_{\varsigma \left(x\right)}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right):=\left\{u\in L^{\mathcal{H}}\left(\mathfrak{Q}\right): | {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }u|\in L^{\mathcal{H}}\left(\mathfrak{Q}\right)\right\},$$

endued with the norm

$${\parallel u\parallel }_{{\mathbb{H}}_{\varsigma \left(x\right)}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right)}={\left|\right|u\left|\right|}_{L^{\mathcal{H}}\left(\mathfrak{Q}\right)}+\left|\right|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }u{\left|\right|}_{L^{\mathcal{H}}\left(\mathfrak{Q}\right)}.$$

Proposition 1

([10]). The space ${\mathbb{H}}_{\varsigma \left(x\right)}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right)$ is a reflexive and separable Banach space.

Remark 2.

We can define $\mathbb{H}\left(\mathfrak{Q}\right):={\mathbb{H}}_{\varsigma \left(x\right),0}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right)$ as the closure of $C_0^{\infty }\left({\mathbb{R}}^N\right)$ in ${\mathbb{H}}_{\varsigma \left(x\right)}^{\gamma ,\beta ,\varphi }\left(\mathfrak{Q}\right)$ which can be reformed by the equivalent norm $\parallel u\parallel :=\parallel {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ,\varphi }{u\parallel }_{L^{\mathcal{H}}\left(\mathfrak{Q}\right)}$. This space is a separable and reflexive Banach space [10].

The results below will be needed for our purposes.

Proposition 2

([10]). $\left(i\right)$$\mathbb{H}\left(\mathfrak{Q}\right)\hookrightarrow L^r\left(\mathfrak{Q}\right)$ is continuous for all $1\le r\le {\varsigma }^{*}:={\displaystyle \frac{N\varsigma }{N-\gamma \varsigma }}$;

$\left(ii\right)$$\mathbb{H}\left(\mathfrak{Q}\right)\hookrightarrow L^r\left(\mathfrak{Q}\right)$ is compact for all $r\in [1,{\varsigma }^{*})$.

Proposition 3

([10]). $\left(i\right)$ If $\mathbf{y}\ne 0$, then ${\parallel \mathbf{y}\parallel }_{\mathcal{H}}=c$ if and only if ${\varrho }^{\mathcal{H}}\left({\displaystyle \frac{\mathbf{y}}{c}}\right)=1$;

(ii) 

${\parallel \mathbf{y}\parallel }_{\mathcal{H}}<1$ (resp. $>1,=1$) if and only if ${\varrho }^{\mathcal{H}}\left(\mathbf{y}\right)<1$ (resp. $>1,=1)$;

(iii) 

If ${\parallel \mathbf{y}\parallel }_{\mathcal{H}}<1$, then ${\parallel \mathbf{y}\parallel }_{\mathcal{H}}^{\alpha }\le {\varrho }^{\mathcal{H}}\left(\mathbf{y}\right)\le {\parallel \mathbf{y}\parallel }_{\mathcal{H}}^{\varsigma }$;

(iv) 

If ${\parallel \mathbf{y}\parallel }_{\mathcal{H}}>1$, then ${\parallel \mathbf{y}\parallel }_{\mathcal{H}}^{\varsigma }\le {\varrho }^{\mathcal{H}}\left(\mathbf{y}\right)\le {\parallel \mathbf{y}\parallel }_{\mathcal{H}}^{\alpha }$;

(v) 

${\parallel \mathbf{y}\parallel }_{\mathcal{H}}\to 0$ if and only if ${\varrho }^{\mathcal{H}}\left(\mathbf{y}\right)\to 0$;

(vi) 

${\parallel \mathbf{y}\parallel }_{\mathcal{H}}\to \infty$ if and only if ${\varrho }^{\mathcal{H}}\left(\mathbf{y}\right)\to \infty$.

3. Main Result

The principle outcome established in this paper is formulated as follows:

Theorem 1.

Assume that the hypotheses $\left(\mathbf{H}\mathbf{1}\right)$ and $\left(\mathbf{H}\mathbf{2}\right)$ are satisfied and $g(x,0)=0$; then, there exists a sequence ${\left\{{\varpi }_k\right\}}_{k\in \mathbb{N}}\subset \mathbb{H}\left(\mathfrak{Q}\right)$ of positive weak solutions of problem (1), where

$$\left[{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}G(x,{\varpi }_k)dx\right]⟶0, \mathrm{as} k⟶+\infty$$

and

$$\parallel {\varpi }_k\parallel ⟶0⟶0, \mathrm{as} k\to +\infty .$$

Definition 1.

We say that $\varpi \in \mathbb{H}\left(\mathfrak{Q}\right)$ is weak solution of (1) if

$${\int }_{\mathfrak{Q}}\left({\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma -2} {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v+\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v\right)dx={\int }_{\mathfrak{Q}}g(x,\varpi )vdx,$$

for all $v\in \mathbb{H}\left(\mathfrak{Q}\right)$.

Let us introduce the energy functional $\mathfrak{E}$: $\mathbb{H}\left(\mathfrak{Q}\right)⟶\mathbb{R}$ associated to problem (1), which is defined as follows:

$$\mathfrak{E}\left(\varpi \right)={\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}G(x,\varpi )dx.$$

(9)

Keep in mind that $\mathfrak{E}\in C^1\left(\mathbb{H}\left(\mathfrak{Q}\right),\mathbb{R}\right)$, and it is noteworthy that the critical points of $\mathfrak{E}$ correspond to weak solutions of (1) and its Gateaux derivative is

$$\begin{array}{cc}\hfill ⟨{\mathfrak{E}}^{\prime }\left(\varpi \right),v⟩={\int }_{\mathfrak{Q}}({\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v+\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha -2} & {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v)dx\hfill \\ \hfill & -{\int }_{\mathfrak{Q}}ug(x,\varpi )dx.\hfill \end{array}$$

(10)

Let $\mathbf{L}:\mathbb{H}\left(\mathfrak{Q}\right)⟶\mathbb{H}{\left(\mathfrak{Q}\right)}^{*}$, defined as follow

$$⟨\mathbf{L}\left(u\right),v⟩={\int }_{\mathfrak{Q}}\left({\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v+\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi {\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }v\right)dx,$$

for all $\varpi ,v\in \mathbb{H}\left(\mathfrak{Q}\right)$ where $⟨·,·⟩$ denotes duality pairing between $\mathbb{H}\left(\mathfrak{Q}\right)$ and its dual space ${\left(\mathbb{H}\left(\mathfrak{Q}\right)\right)}^{*}$.

Proposition 4

([19]). If $\mathbf{L}$ is as above, then

(i) 

$\mathbf{L}:\mathbb{H}\left(\mathfrak{Q}\right)⟶{\left(\mathbb{H}\left(\mathfrak{Q}\right)\right)}^{*}$ is a continuous, bounded and strictly monotone operator;

(ii) 

$\mathbf{L}:\mathbb{H}\left(\mathfrak{Q}\right)⟶{\left(\mathbb{H}\left(\mathfrak{Q}\right)\right)}^{*}$ is a mapping of type ${\left(S\right)}_{+}$;

(iii) 

$\mathbf{L}:\mathbb{H}\left(\mathfrak{Q}\right)⟶{\left(\mathbb{H}\left(\mathfrak{Q}\right)\right)}^{*}$ is a homeomorphism.

We will proceed to prove Theorem 1 by leveraging an idea originally introduced by Kristǎly, Moroşanu, and Tersian [20], who demonstrated the existence of infinitely many homoclinic solutions for a $\varsigma$-Laplace equation. Initially, based on our hypotheses, concerning g, there exist $z_0>0$ and $t_0>0$ such that $\left|g(x,\varpi )\right|\le z_0$ holds for every $\varpi \in \left[0,{\varpi }_0\right]$ and almost every $x\in \mathfrak{Q}$. Without losing generality, we assume that, for every $k\in \mathbb{N}$, $v_k\le {\varpi }_0$, where $v_k$ is derived from $\left(\mathbf{H}\mathbf{1}\right)$. Let us define

$$\tilde{g}(x,\varpi )=\left\{\begin{array}{cc}0,\hfill & \mathrm{if} \varpi \le 0,\hfill \\ g(x,\varpi ),\hfill & \mathrm{if} 0<\varpi \le {\varpi }_0,\hfill \\ g\left(x,{\varpi }_0\right),\hfill & \mathrm{if} \varpi >{\varpi }_0.\hfill \end{array}\right.$$

(11)

Therefore, one has

$$|\tilde{g}(x,\varpi )|\le z_0, \forall \varpi \in \mathbb{R} \mathrm{and} \mathrm{a}.\mathrm{e}. x\in \mathfrak{Q}.$$

(12)

Let us now consider the following problem:

$$\left\{\begin{array}{cc}{\mathbb{D}}_T^{\gamma ,\beta ;\varphi }\left({\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi +\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha -2}{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right)=\tilde{g}(x,\varpi ),\hfill & \mathrm{in} \mathfrak{Q},\hfill \\ \hfill \\ \varpi =0,\hfill & \mathrm{on} \partial \mathfrak{Q}.\hfill \end{array}\right.$$

(13)

Therefore, the weak solutions of problem (13) correspond to the critical points of the functional

$$\tilde{\mathfrak{E}}\left(\varpi \right)={\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}\tilde{G}(x,\varpi )dx,$$

(14)

where ${\displaystyle \tilde{G}(x,\varpi )={\int }_0^{\varpi }\tilde{g}(x,s)ds}$.

Remark 3.

Given (12), it becomes apparent that $\tilde{\mathfrak{E}}$ is validly defined, demonstrates weak sequential lower semi-continuity, and possesses Gateaux differentiable within $\mathbb{H}\left(\mathfrak{Q}\right)$.

For every fixed $k\in N$, let us define the set

$${\mathbb{S}}_k=\left\{\varpi \in \mathbb{H}\left(\mathfrak{Q}\right):\varpi \left(x\right)\ne 0 \mathrm{and} 0\le \varpi \left(x\right)\le v_k \mathrm{a}.\mathrm{e}. x\in \mathfrak{Q}\right\}.$$

Lemma 1.

The functional $\tilde{\mathfrak{E}}$ is bounded from below on ${\mathbb{S}}_k$ and ${\displaystyle {\rho }_k:=\underset{\varpi \in {\mathbb{S}}_k}{inf}\tilde{\mathfrak{E}}\left(\varpi \right)}$ is attained at ${\varpi }_k\in {\mathbb{S}}_k$.

Proof. 

For every $k\in \mathbb{N}$ and using (12), we acquire, for every $\varpi \in {\mathbb{S}}_k$,

$$\begin{array}{cc}\hfill \tilde{\mathfrak{E}}\left(\varpi \right)& =\underset{>0}{\underbrace{{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx}}+\underset{>0}{\underbrace{{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx}}-{\int }_{\mathfrak{Q}}\tilde{G}(x,\varpi )dx\hfill \\ \hfill & \ge -{\int }_{\mathfrak{Q}}\tilde{G}(x,u)dx\hfill \\ \hfill & \ge -z_0{v}_k\left|\mathfrak{Q}\right|.\hfill \end{array}$$

(15)

This implies that $\tilde{\mathfrak{E}}$ is bounded from below on ${\mathbb{S}}_k$. Furthermore, it is evident that ${\mathbb{S}}_k$ is closed and convex, hence weakly closed in $\mathbb{H}\left(\mathfrak{Q}\right)$. Let ${\left\{{\varpi }_n\right\}}_{n\in \mathbb{N}}\subset {\mathbb{S}}_k$ such that

$${\rho }_k\le \tilde{\mathfrak{E}}\left({\varpi }_n\right)\le {\rho }_k+{\displaystyle \frac{1}{n}}, \mathrm{for} \mathrm{all} n\in \mathbb{N}.$$

If $∥{\varpi }_n∥\le 1$, then we are finished, otherwise, we have

$${\displaystyle \frac{1}{\alpha }}{∥{\varpi }_n∥}^{\varsigma }\le {\rho }_k+1+z_0{v}_k\left|\mathfrak{Q}\right|, \mathrm{for} \mathrm{all} n\in \mathbb{N}.$$

(16)

Thus, we conclude that ${\left\{{\varpi }_n\right\}}_{n\in \mathbb{N}}$ is bounded in $\mathbb{H}\left(\mathfrak{Q}\right)$. Thus, by taking a sub-sequence, ${\varpi }_n\rightharpoonup {\varpi }_k\in {\mathbb{S}}_k$. Consequently, due to the weak sequential lower semi-continuity of $\tilde{\mathfrak{E}}$, we achieve $\tilde{\mathfrak{E}}\left({\varpi }_k\right)={\rho }_k$. □

Lemma 2.

For every $k\in \mathbb{N}$, $0\le {\varpi }_k\left(x\right)\le u_k$ a.e. $x\in \mathfrak{Q}$.

Proof. 

We defined $\mathrm{{\rm Y}}=\left\{x\in \mathfrak{Q}:u_k<{\varpi }_k\left(x\right)\le v_k\right\}$ and we suppose that $meas\left(\mathrm{{\rm Y}}\right)>0$. Let us consider the function $\mathfrak{h}\left(t\right)=min\left\{t^{+},u_k\right\}$ and $w_k=\mathfrak{h}\left({\varpi }_k\right)$, where $t^{+}=max\{0,t\}$. Clearly, $\mathfrak{h}$ is continuous in $\mathbb{H}\left(\mathfrak{Q}\right)$. Additionally, we have $0\le w_k\left(x\right)\le u_k$ for a.e. $x\in \mathfrak{Q}$. Hence, $w_k\in {\mathbb{S}}_k$ and

$$w_k\left(x\right)=\left\{\begin{array}{cc}{\varpi }_k\left(x\right),\hfill & \mathrm{if} x\in \mathfrak{Q}\setminus \mathrm{{\rm Y}},\hfill \\ u_k,\hfill & \mathrm{if} x\in \mathrm{{\rm Y}}.\hfill \end{array}\right.$$

(17)

Furthermore, it follows that

$$\begin{array}{cc}\hfill \tilde{\mathfrak{E}}\left(w_k\right)-\tilde{\mathfrak{E}}\left({\varpi }_k\right)=& {\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{w}_k\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{w}_k\right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}\tilde{G}(x,w_k)dx\hfill \\ & -{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx-{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx+{\int }_{\mathfrak{Q}}\tilde{G}(x,{\varpi }_k)dx\hfill \\ \hfill =& {\displaystyle \frac{1}{\varsigma }}{\int }_{\mathrm{{\rm Y}}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{w}_k\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathrm{{\rm Y}}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{w}_k\right|}^{\alpha }dx-{\int }_{\mathrm{{\rm Y}}}\tilde{G}(x,w_k)dx\hfill \\ & -{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathrm{{\rm Y}}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx-{\displaystyle \frac{1}{\alpha }}{\int }_{\mathrm{{\rm Y}}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx+{\int }_{\mathrm{{\rm Y}}}\tilde{G}(x,{\varpi }_k)dx\hfill \\ \hfill =& -{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathrm{{\rm Y}}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx-{\displaystyle \frac{1}{\alpha }}{\int }_{\mathrm{{\rm Y}}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx\hfill \\ & -{\int }_T\underset{\ge 0}{\underbrace{\left(\tilde{G}\left(x,u_k\right)-\tilde{G}\left(x,{\varpi }_k\right)\right)}}dx.\hfill \end{array}$$

Therefore, $\tilde{\mathfrak{E}}\left(w_k\right)-\tilde{\mathfrak{E}}\left({\varpi }_k\right)\le 0$. On the other hand, since $w_k\in {\mathbb{S}}_k$,

$$\tilde{\mathfrak{E}}\left(w_k\right)\ge \tilde{\mathfrak{E}}\left({\varpi }_k\right)=\underset{w\in {\mathbb{S}}_k}{inf}\tilde{\mathfrak{E}}\left(w\right).$$

Thus, each term in $\tilde{\mathfrak{E}}\left(w_k\right)-\tilde{\mathfrak{E}}\left({\varpi }_k\right)$ must be zero. Particularly,

$${\displaystyle \frac{1}{\varsigma }}{\int }_{\mathrm{{\rm Y}}}|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k{|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathrm{{\rm Y}}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx={\int }_T\left(\tilde{G}\left(x,u_k\right)-\tilde{G}\left(x,{\varpi }_k\right)\right)dx=0,$$

implying that $meas\left(\mathrm{{\rm Y}}\right)=0$. □

Lemma 3.

For each $k\in \mathbb{N}$, ${\varpi }_k$ stands as a local minimum point of $\tilde{\mathfrak{E}}$ within $\mathbb{H}\left(\mathfrak{Q}\right)$.

Proof. 

Consider ${\mathrm{{\rm Y}}}^{\prime }=\left\{x\in \mathfrak{Q}:\varpi \left(x\right)\notin (0,u_k]\right\}$. Let $w=h\left(\varpi \right)$; then, we can see that

$${\int }_w^{\varpi }\tilde{g}(x,s)ds=0, \mathrm{for} \mathrm{all} x\in \mathfrak{Q}\setminus {\mathrm{{\rm Y}}}^{\prime }.$$

Furthermore, when $x\in {\mathrm{{\rm Y}}}^{\prime }$, the situation can be categorized into the following three cases:

• Case 1: When $\varpi \left(x\right)<0$, then, we have ${\displaystyle {\int }_w^{\varpi }\tilde{g}(x,s)ds=0.}$
• Case 2: When $u_k<\varpi \left(x\right)\le v_k$, using $\left(\mathbf{H}\mathbf{1}\right)$, we obtain that ${\displaystyle {\int }_w^{\varpi }\tilde{g}(x,s)ds\le 0}$.
• Case 3: When $\varpi \left(x\right)>v_k$, then, we have

  $${\displaystyle {\int }_w^{\varpi }\tilde{g}(x,s)ds={\int }_{u_k}^{\varpi }\tilde{g}(x,s)ds\le {\int }_{u_k}^{\varpi }{z}_{0}ds=z_0\left(\varpi \left(x\right)-u_k\right).}$$

Now, consider ${\varsigma }^{*}\ge \xi +1>\alpha$ for every $x\in \mathfrak{Q}$, and let us fix it. Then, the constant

$${\lambda }_0:=\underset{s\ge v_k}{sup}{\displaystyle \frac{z_0\left(s-u_k\right)}{{\left(s-u_k\right)}^{\xi +1}}},$$

is finite. For a.e. $x\in \mathfrak{Q}$, we have

$${\int }_w^{\varpi }\tilde{g}(x,s)ds\le {\lambda }_0{|\varpi \left(x\right)-w\left(x\right)|}^{\xi +1}.$$

Thus, by employing Proposition 2, we obtain

$${\int }_{\mathfrak{Q}}{\int }_w^{\varpi }\tilde{g}(x,s)dsdx\le {\lambda }_0{C}^{\xi +1}{\parallel \varpi -w\parallel }^{\xi +1},$$

(18)

where C is the embedding constant of $\mathbb{H}\left(\mathfrak{Q}\right)\hookrightarrow L^{\xi +1}\left(\mathfrak{Q}\right)$. Therefore, by using (18), one has

$$\begin{array}{cc}\hfill \tilde{\mathfrak{E}}\left(\varpi \right)-\tilde{\mathfrak{E}}\left(w\right)& ={\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}G(x,\varpi )dx\hfill \\ \hfill & -{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }w\right|}^{\varsigma }dx-{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }w\right|}^{\alpha }dx+{\int }_{\mathfrak{Q}}G(x,w)dx\hfill \\ \hfill & ={\displaystyle \frac{1}{\varsigma }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx-{\int }_{{\mathrm{{\rm Y}}}^{\prime }}G(x,\varpi )dx\hfill \\ \hfill & -{\displaystyle \frac{1}{\varsigma }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }w\right|}^{\varsigma }dx-{\displaystyle \frac{1}{\alpha }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }w\right|}^{\alpha }dx+{\int }_{{\mathrm{{\rm Y}}}^{\prime }}G(x,w)dx\hfill \\ \hfill & ={\displaystyle \frac{1}{\varsigma }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{{\mathrm{{\rm Y}}}^{\prime }}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }\varpi \right|}^{\alpha }dx-{\int }_{{\mathrm{{\rm Y}}}^{\prime }}{\int }_w^{\varpi }\tilde{g}(x,s)dsdx\hfill \\ \hfill & ={\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }(\varpi -w)\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }(\varpi -w)\right|}^{\alpha }dx-{\int }_{{\mathrm{{\rm Y}}}^{\prime }}{\int }_{u_k}^{\varpi }g(x,s)dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }(\varpi -w)\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }(\varpi -w)\right|}^{\alpha }dx-{\lambda }_0{C}^{\xi +1}{\parallel \varpi -w\parallel }^{\xi +1}.\hfill \end{array}$$

(19)

However, considering $w\in {\mathbb{S}}_k$, we deduce

$$\tilde{\mathfrak{E}}\left(w\right)\ge \tilde{\mathfrak{E}}\left({\varpi }_k\right).$$

(20)

Utilizing Proposition 3 and (20) in (19), we derive

$$\begin{array}{cc}\hfill \tilde{\mathfrak{E}}\left(\varpi \right)\ge & \tilde{\mathfrak{E}}\left({\varpi }_k\right)+{\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{(\varpi -w)|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }(\varpi -w)\right|}^{\alpha }dx\hfill \\ & -{\lambda }_0{C}^{\xi +1}{\parallel \varpi -w\parallel }^{\xi +1}\hfill \\ \hfill \ge & \tilde{\mathfrak{E}}\left({\varpi }_k\right)+{\displaystyle \frac{1}{\alpha }}{\parallel \varpi -w\parallel }^{\tau }-{\lambda }_0{C}^{\xi +1}{\parallel \varpi -w\parallel }^{\xi +1}\hfill \\ \hfill =& \tilde{\mathfrak{E}}\left({\varpi }_k\right)+{\parallel \varpi -w\parallel }^{\tau }\left({\displaystyle \frac{1}{\alpha }}-{\lambda }_0{C}^{\xi +1}{\parallel \varpi -w\parallel }^{\xi +1-\tau }\right),\hfill \end{array}$$

where $\tau =\varsigma$ when $\parallel \varpi -w\parallel \ge 1$ and $\tau =\alpha$ when $\parallel \varpi -w\parallel \le 1$. As $\mathfrak{h}$ is continuous, there exists $\delta >0$ such that, for every $\varpi \in \mathbb{H}\left(\mathfrak{Q}\right)$ with

$$∥\varpi -{\varpi }_k∥<\delta \mathrm{and} \parallel \varpi -w\parallel \le {\displaystyle \frac{1}{\alpha {\lambda }_0{C}^{\xi +1}}}.$$

This implies that that ${\varpi }_k$ is a local minimum of $\tilde{\mathfrak{E}}$. □

Lemma 4.

For every $k\in \mathbb{N}$, we have ${\rho }_k<0$ and ${\displaystyle \underset{k\to +\infty }{lim}{\rho }_k=0}$.

Proof. 

Considering $\left(\mathbf{H}\mathbf{2}\right)$, it is straightforward to deduce that $z_k\in {\mathbb{S}}_k$. Consequently,

$$\begin{array}{cc}\hfill {\rho }_k\le \tilde{\mathfrak{E}}\left(z_k\right)& =-{\int }_{\mathfrak{Q}}\tilde{G}\left(x,z_k\right)dx\hfill \\ \hfill & =-{\int }_{\mathfrak{Q}}G\left(x,z_k\right)dx\hfill \\ \hfill & =-{\int }_{\mathfrak{Q}}{\int }_0^{z_k}g\left(x,z_k\right)dx<0.\hfill \end{array}$$

(21)

Next, we will demonstrate that ${\displaystyle \underset{k\to +\infty }{lim}{\rho }_k=0}$. Following Lemma 1, for each $k\in \mathbb{N}$ and ${\varpi }_k\in {\mathbb{S}}_k$, we acquire

$$\begin{array}{cc}\hfill {\rho }_k& =\tilde{\mathfrak{E}}\left({\varpi }_k\right)\hfill \\ & ={\displaystyle \frac{1}{\varsigma }}{\int }_{\mathfrak{Q}}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }dx-{\int }_{\mathfrak{Q}}\tilde{G}\left(x,{\varpi }_k\right)d x\hfill \\ & \ge -z_{0}meas\left(\mathfrak{Q}\right)v_k.\hfill \end{array}$$

As ${\displaystyle \underset{k\to +\infty }{lim}{v}_k=0}$, we obtain ${\displaystyle \underset{k\to +\infty }{lim}{\rho }_k\ge 0}$. Observing that ${\rho }_k<0$, it follows that ${\displaystyle \underset{k\to +\infty }{lim}{\rho }_k=0}$. □

Now, let us prove our main result.

Proof of Theorem 1.

Given that ${\varpi }_k$ are local minima of $\tilde{\mathfrak{E}}$, they serve as critical points of $\tilde{\mathfrak{E}}$, hence weak solutions of (1). Referring to Lemma 2, we can infer the existence of infinitely many distinct ${\varpi }_k$ with ${\displaystyle \underset{k\to +\infty }{lim}{∥{\varpi }_k∥}_{\infty }=0}$. Furthermore, we have

$$\begin{array}{cc}\hfill {∥{\varpi }_k∥}^{\tau }& \le {\displaystyle \frac{1}{\varsigma }}{\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\varsigma }dx+{\displaystyle \frac{1}{\alpha }}{\int }_{\mathfrak{Q}}\mathbf{b}\left(x\right){\left|{\mathbb{D}}_{0^{+}}^{\gamma ,\beta ;\varphi }{\varpi }_k\right|}^{\alpha }d x\hfill \\ & ={\rho }_k+{\int }_{\mathfrak{Q}}\tilde{G}\left(x,{\varpi }_k\right)d x\hfill \\ & \le {\rho }_k+meas\left(\mathfrak{Q}\right)z_0{v}_k,\hfill \end{array}$$

where $\tau =\varsigma$ when $∥{\varpi }_k∥\ge 1$ and $\tau =\alpha$ when $∥{\varpi }_k∥\le 1$. Hence, we deduce that ${\displaystyle \underset{k\to +\infty }{lim}\left|{\varpi }_k\right|=0}$, thereby concluding the proof. □

4. Conclusions

As a conclusion, in our study, we investigated a novel class of double-phase problems involving the $\varsigma$-Laplacian operator and $\varphi$-Hilfer fractional derivatives. Using variational techniques and weighted Musielak space theory, we established the existence of infinitely many positive solutions under suitable assumptions on the nonlinearities. Our results significantly advance the understanding of such problems and provide a foundation for further research in fractional calculus and its applications. The result has potential applications in several fields such as non-Newtonian fluids, diffusion processes, and mathematical modeling, contributing to both theoretical and applied mathematics.

In the future, we will study the following:

1.

Addressing the digital aspect of this paper to study a concrete model that simplifies the original ones.

2.

Problems involving variable exponents and singular nonlinearities.