A critical nonlinear elliptic equation with non local regional diffusion

Abstract

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent \begin{eqnarray*} &ε^{2α} (-Δ)_ρ^αu + u = λu^q + u^{2_α^{*}-1} \mbox{ in } \mathbb{R}^{N}, \\ & u \in H^α(\mathbb{R}^{N}), \end{eqnarray*} where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q\in (1,2_{\alpha }^{\ast }-1)$, $2_{\alpha }^{\ast }=\frac{2N}{N-2\alpha }$ is the critical Sobolev exponent, $\lambda >0$ is a parameter and $(-\Delta {)}_{\rho }^{\alpha }$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $\rho (x)>0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\epsilon \to 0$.