A critical nonlinear elliptic equation with non local regional diffusion

Abstract

In this article we are interested in the nonlocal regional Schr\"odinger equation with critical exponent \begin{eqnarray*} &\epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha}u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \mbox{ in } \mathbb{R}^{N}, \\ & u \in H^{\alpha}(\mathbb{R}^{N}), \end{eqnarray*} where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q\in (1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*} = \frac{2N}{N-2\alpha}$ 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 $\varepsilon\to 0$.