Existence and Concentration of Positive Solutions for the Planar Choquard-Type Double-Phase Problems With Critical Exponential Growth

Abstract

In this article, we consider the following double-phase problems with Choquard-type nonlinearity: $$\begin{array}{r}\{\begin{array}{l}-{ϵ}^p{\mathrm{\Delta }}_{p}v-{ϵ}^2\mathrm{\Delta }v+V(x)(v+|v{|}^{p-2}v)=f(v)+(I_{\alpha }*|v{|}^2)v\text{in}{\mathbb{R}}^2,\\ v>0,v\in W^{1,p}({\mathbb{R}}^2)\cap W^{1,2}({\mathbb{R}}^2),\end{array}\end{array}$$ where $1<p<2$ , $ϵ>0$ is a parameter, $V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R})$ , $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with critical exponential growth condition, $I_{\alpha }$ is the Riesz potential with $0<\alpha <2$ . Under some hypotheses, the existence and concentration behavior of positive solutions are established by variational methods.