Multibumpsolutions of nonlinear Schrödinger equations with steep potential welland indefinite potential

Thomas Bartsch; Zhongwei Tang

doi:10.3934/dcds.2013.33.7

Article Contents

2013, Volume 33, Issue 1: 7-26. Doi: 10.3934/dcds.2013.33.7

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Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential

Thomas Bartsch^1, and
Zhongwei Tang^2,

1.
Mathematisches Institut, University of Giessen, Arndtstr. 2 35392 Giessen
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: July 31, 2011

Revised: November 30, 2011

Published: December 2012

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Abstract

We are concerned with the existence of single- and multi-bump solutions of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$; here $p>2$, and $p<\frac{2N}{N-2}$ if $N\geq 3$. We require that $a\geq 0$ is in $L^\infty_{loc}({\mathbb R}^N)$ and has a bounded potential well $\Omega$, i.e. $a(x)=0$ for $x\in\Omega$ and $a(x)>0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator $L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of $u_\lambda$ may be sign-changing.

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Mathematics Subject Classification: 35J61, 35J91, 35J20, 35B40, 47J30.

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References

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