On existence and concentration behavior of positive ground state solutions for a class of fractional Schrödinger–Choquard equations

Abstract

In this paper, we first employ variational methods to show the existence of positive ground state solutions for fractional Schrödinger–Choquard equations

$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^{s}u+Vu=(I_{\alpha }*|u|^p)|u|^{p-2}u,~~~\mathrm {in}~~~\mathbb {R}^{N}, \end{aligned}$$

where potential $V(x)\in C(\mathbb {R}^{N})$ is nonnegative and bounded away from 0 as $|x|\rightarrow \infty$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$ and $\varepsilon >0$ is a parameter small enough. When $V(x)\in C(\mathbb {R}^{N})$ achieves 0 with a homogeneous behavior, we then investigate the concentration behavior of positive ground state solutions as $\varepsilon \rightarrow 0^{+}$.