Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

Wei  Long; Shuangjie Peng; Jing Yang

doi:10.3934/dcds.2016.36.917

Article Contents

2016, Volume 36, Issue 2: 917-939. Doi: 10.3934/dcds.2016.36.917

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Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079
2.
Department of Mathematics, Huazhong Normal University,Wuhan, 430079

Received: February 28, 2014

Revised: December 31, 2014

Published: May 2017

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Abstract

We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.

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Mathematics Subject Classification: 35J20, 35J60.

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