July  2008, 7(4): 883-903. doi: 10.3934/cpaa.2008.7.883

Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency

1. 

Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received  June 2007 Revised  January 2008 Published  April 2008

We study the nonlinear Schrödinger equations:

$-\epsilon^2 \Delta u+V(x)u=f(u), \quad u\in H^1(\mathbb R^N),$ $\qquad\qquad\qquad$ (*)$_\epsilon $

where $V$ satisfies $ V(x)\geq 0 $, liminf$_{ |x|\to \infty }V(x)>0$. $f(u)\in C^1(\mathbb R,\mathbb R)$ satisfies Ambrosetti--Rabinowitz condition and some properties and $\frac{f(u)}{u}$ is nondecreasing. We consider the case inf$_{x\in \mathbb R^N}V(x)>0$ and critical frequency case, that is, inf$_{x\in \mathbb R^N}V(x)=0$. We study the existence of sign-changing 2-peak solutions of (*)$_\epsilon$ whose one peak is positive, another peak is negative and both peaks concentrate to a same local minimum point of $V(x)$ as $\epsilon \to 0$.

Citation: Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883
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