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 $\epsilon \to 0$.
Categories: math.AP