Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems

Abstract

We consider the following singularly perturbed nonlocal elliptic problem

$$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=\displaystyle \varepsilon ^{\alpha -3}(W_{\alpha }(x)*|u|^{p})|u|^{p-2}u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$

where $\varepsilon >0$ is a parameter, $a>0,b\ge 0$ are constants, $\alpha \in (0,3)$, $p\in [2, 6-\alpha )$, $W_{\alpha }(x)$ is a convolution kernel and V(x) is an external potential satisfying some conditions. By using variational methods, we establish the existence and concentration of positive ground state solutions for the above equation.