Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation

Patricio Felmer; César Torres

doi:10.3934/cpaa.2014.13.2395

Article Contents

2014, Volume 13, Issue 6: 2395-2406. Doi: 10.3934/cpaa.2014.13.2395

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Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation

Patricio Felmer^1, and
César Torres^2,

1.
Departamento de Ingeneria Matematica F.C.F.M., Universidad de Chile, Casilla 170 Correro 3, Santiago
2.
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago

Received: November 2013

Revised: March 2014

Published: November 2014

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Abstract

The aim of this paper is to study radial symmetry properties for ground state solutions of elliptic equations involving a regional fractional Laplacian, namely \begin{eqnarray} (-\Delta)_{\rho}^{\alpha}u + u = f(u) \quad \mbox{in} \ \mathbb{R}^{n}, \ \ \mbox{for} \ \ \alpha\in (0,1). \end{eqnarray} In [9], the authors proved that problem (1) has a ground state solution. In this work we prove that the ground state level is achieved by a radially symmetry solution. The proof is carried out by using variational methods jointly with rearrangement arguments.

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Mathematics Subject Classification: Primary: 35J60, 35R11; Secondary: 35J20, 35B07.

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