Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces

In this study, we examine a particular type of fractional \((\phi _1,\phi _2)\) $({\varphi }_1,{\varphi }_2)$ -Laplacian problem, which can be represented by the following equation: \(\begin{aligned} \displaystyle {\left\{ \begin{array}{ll} (-\Delta )^{\textrm{s}}_{\phi _1(\cdot )}\hbox {u}+(-\Delta )^{\textrm{s}}_{\phi _{2}(\cdot )}\hbox {u}+\hbox {V}(\epsilon \hbox {x})(\phi _1(|\hbox {u}|)\hbox {u}+\phi _2(|\hbox {u}|)\hbox {u})=\mathrm{f(x,u)} &{} \text{ in } \mathbb {R}^{\textrm{n}},\\ u\in W_\epsilon ,\quad u>0\quad \text{ in } \mathbb {R}^{\textrm{n}}. \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle \left\{\begin{array}{cc}{(-\mathrm{\Delta })}_{{\varphi }_1(·)}^{\text{s}}\text{u}+{(-\mathrm{\Delta })}_{{\varphi }_2(·)}^{\text{s}}\text{u}+\text{V}(ϵ\text{x})({\varphi }_1(|\text{u}|)\text{u}+{\varphi }_2(|\text{u}|)\text{u})=\mathrm{f}(\mathrm{x},\mathrm{u})\hfill & \text{in}{\mathbb{R}}^{\text{n}},\hfill \\ u\in W_{ϵ},u>0\text{in}{\mathbb{R}}^{\text{n}}.\hfill \end{array}\right)}\end{array}$$ Here, \(s\in (0, 1)\) $s\in (0,1)$ , \(\epsilon > 0\) $ϵ>0$ , and \((-\Delta )^s_{\phi _i(.)}\) ${(-\mathrm{\Delta })}_{{\varphi }_i(.)}^s$ (where \(i=1,2\) $i=1,2$ ) is a fractional \(\phi _i\) ${\varphi }_i$ -Laplacian operator. The potential function \(V:\mathbb {R}^n\rightarrow \mathbb {R}\) $V:{\mathbb{R}}^n\to \mathbb{R}$ is a continuous, possibly unbounded, function and \(f:\mathbb {R}\rightarrow \mathbb {R}\) $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinearity with Orlicz subcritical growth. Our goal is to investigate the multiplicity and concentration properties of the solutions to this problem as \(\epsilon \) $ϵ$ approaches zero, using the Nehari manifold and the penalization method as our primary tools.