Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

Patricio Felmera1, Alexander Quaasa2 and Jinggang Tana2

a1 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile (pfelmer@dim.uchile.cl)

a2 Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Avenida España 1680, Valparaíso, Chile alexander.quaas@usm.cl; jinggang.tan@usm.cl)

Abstract

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional Laplacian

\begin{gather*}(-\Delta)^{\alpha}u+u=f(x,u)\text{~in~}\mathbb{R}^{N},\\u>0\text{~in~}\mathbb{R}^{N},\qquad\lim_{|x|\rightarrow\infty}u(x)=0.\end{gather*}

Furthermore, we analyse the regularity, decay and symmetry properties of these solutions.

(Received April 28 2011)

(Accepted October 27 2011)