Existence and concentration of solution for a non-local regional Schr\"odinger equation with competing potentials

Abstract

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schr\"odinger equation $$ \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. $$ where $\epsilon$ is a positive parameter, $0< \alpha < 1$, $1<p<\frac{n+2\alpha}{n-2\alpha}$, $n>2\alpha$; $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius $\rho (x)>0$, $\rho, Q, K$ are competing functions. We study the existence of ground state and we analyze the behavior of semi-classical solutions as $\epsilon \to 0$.