A Critical Nonlinear Elliptic Equation with Nonlocal Regional Diffusion

Abstract

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent \[ \epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha} u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \quad \textrm{in $\mathbb{R}^{n}$}, \quad u \in H^{\alpha}(\mathbb{R}^{n}), \] where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q \in (1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*} = 2n/(n-2\alpha)$ is the critical Sobolev exponent, $\lambda \gt 0$ is a parameter and $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional Laplacian, whose range of scope is a ball with radius $\rho(x) \gt 0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon \to 0$.