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NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  10 October 2014

X. H. TANG
X. H. TANG*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China email tangxh@csu.edu.cn
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Abstract

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We consider the semilinear Schrödinger equation

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-\triangle u+V(x)u=f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\end{array}\right.\end{eqnarray}$$
where $f(x,u)$ is asymptotically linear with respect to $u$, $V(x)$ is 1-periodic in each of $x_{1},x_{2},\dots ,x_{N}$ and $\sup [{\it\sigma}(-\triangle +V)\cap (-\infty ,0)]<0<\inf [{\it\sigma}(-\triangle +V)\cap (0,\infty )]$. We develop a direct approach to find ground state solutions of Nehari–Pankov type for the above problem. The main idea is to find a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold ${\mathcal{N}}^{-}$ by using the diagonal method.

Keywords

Schrödinger equationasymptotically linearnon-Nehari-manifold methodground state solutions of Nehari–Pankov type

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 104 - 116
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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