Non-linear Schrödinger equation with non-local regional diffusion

Article

Abstract

In this article we are interested in the nonlinear Schrödinger equation with non-local regional difussion
$$\begin{aligned}&\epsilon ^{2\alpha } (-\Delta )_{\rho }^{\alpha }u + u = f(u) \hbox { in } \mathbb {R}^{n}, \\&u \in H^{\alpha }(\mathbb {R}^{n}), \end{aligned}$$
where \(f\) is a super-linear sub-critical function 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 \(\varepsilon \rightarrow 0\).

Mathematics Subject Classification

45G05 35J60 35B25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departamento de Ingeniería Matemática and Centro de Modelamiento, Matemático UMR2071 CNRS-UChileUniversidad de ChileSantiagoChile

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