Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth

This article deals with the fractional Choquard equation involving exponential growth as follows \( \varepsilon ^{N}(-\Delta )_{p}^{s}u+Z(x)|u|^{p-2}u =\varepsilon ^{\mu -N}\left[ \frac{1}{|x|^{\mu }}*H(u)\right] h(u)\; \ \ \text {in}\; \mathbb R^{N},\) $${\epsilon }^N{(-\mathrm{\Delta })}_p^{s}u+Z(x){|u|}^{p-2}u={\epsilon }^{\mu -N}\left[\frac{1}{{|x|}^{\mu }},,\ast ,H,(u)\right]h(u)\text{in}{\mathbb{R}}^N,$$ where \(\varepsilon \) $\epsilon$ a positive parameter, \(s\in (0,1)\) $s\in (0,1)$ , \(\mu \in (0, N)\) $\mu \in (0,N)$ and \((-\Delta )_{p}^{s}\) ${(-\mathrm{\Delta })}_p^s$ is a fractional p-Laplace operator with \(p=\frac{N}{s}\ge 2.\) $p=\frac{N}{s}\ge 2.$ The function h is only continuous and has exponential growth. In addition, the potential function Z satisfies some appropriate conditions. We use the Trudinger–Moser inequality to deal with the function h involving exponential growth. Together with the Ljusternik–Schnirelmann category theory and variational method, the multiplicity and concentration behavior of positive solutions are obtained for the above problem. As far as we know, our results seem to be new for the fractional \(\frac{N}{s}\) $\frac{N}{s}$ -Laplacian Choquard equation.