Existence and multiplicity of solutions for a nonlinear Schrödinger equation with non-local regional diffusion

Abstract

In this article we are interested in the following non-linear Schrödinger equation with non-local regional diffusion $$(-\Delta {)}_{{\rho }_{\epsilon }}^{\alpha }u+u=f(u)in{\mathbb{R}}^n,u\in H^{\alpha }({\mathbb{R}}^n),(P_{\epsilon })$$ where $\epsilon >0$, $0<\alpha <1$, $(-\Delta {)}_{{\rho }_{\epsilon }}^{\alpha }$ is a variational version of the regional laplacian, whose range of scope is a ball with radius ${\rho }_{\epsilon }(x)=\rho (\epsilon x)>0$, where $\rho$ is a continuous function. We give general conditions on $\rho$ and $f$ which assure the existence and multiplicity of solution for $(P_{\epsilon })$.

arXiv admin note: text overlap with arXiv:1311.0708 by other authors

References in corpus (1)

• The Dirichlet elliptic problem involving regional fractional Laplacian