Existence and Concentration of Solutions to a Choquard Equation Involving Fractional p-Laplace via Penalization Method

In this paper, we study the Choquard equation involving \(\frac{N}{s}\) $\frac{N}{s}$ -fractional Laplace as follows: \(\begin{aligned} \varepsilon ^{ps}(-\Delta )_{p}^{s}u+V(x)|u|^{p-2}u= \varepsilon ^{\mu -N}\left[ \dfrac{1}{|x|^{\mu }}*F(u)\right] f(u)\;\text {in}\; \mathbb {R}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill {\epsilon }^{\mathit{ps}}{(-\mathrm{\Delta })}_p^{s}u+V(x){|u|}^{p-2}u={\epsilon }^{\mu -N}\left[{\displaystyle \frac{1}{{|x|}^{\mu }}},,\ast ,F,(u)\right]f(u)\text{in}{\mathbb{R}}^N,\end{array}$$ where \(\varepsilon \) $\epsilon$ is a positive parameter, \(N=ps, s\in (0,1), 0<\mu <ps\) $N=ps,s\in (0,1),0<\mu <ps$ . The nonlinear function f and potential function V are continuous and satisfy some suitable conditions. We prove the existence, multiplicity, and concentration of solutions of the above equation by penalization method, Nehari manifold, and Ljusternik-Schnirelmann theory.