Existence and concentration of solution for a non-local regional Schrödinger equation with competing potentials

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

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