Nodal solutions for nonlinear Schrödinger equations with decaying potential

Zuji  Guo

doi:10.3934/cpaa.2016.15.1125

Article Contents

2016, Volume 15, Issue 4: 1125-1138. Doi: 10.3934/cpaa.2016.15.1125

This issue Previous Article Nonlinear noncoercive Neumann problems Next Article Existence and uniqueness for $\mathbb{D}$-solutions of reflected BSDEs with two barriers without Mokobodzki's condition

Nodal solutions for nonlinear Schrödinger equations with decaying potential

Zuji Guo^1,

1.
School of Mathematics, Taiyuan University of Technology, Taiyuan 030024

Received: October 31, 2014

Revised: May 31, 2015

Published: June 2016

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Abstract

This paper concerns the following nonlinear Schrödinger equations: \begin{eqnarray} \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\Delta u +V(x)u= |u|^{p_+-2}u^++|u|^{p_--2}u^-,\ x\in\mathbb{R}^N,\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0, \\ \end{array} \right. \end{eqnarray} where $N\geq 3$ and $2 < p_{\pm} < \frac{2N}{N-2}$. We obtain nodal solutions for the above nonlinear Schrödinger equations with decaying and vanishing potential at infinity, i.e., $\lim\limits_{|x|\rightarrow\infty}V(x)=0$.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J65.

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