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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells


Authors: Yohei Sato and Kazunaga Tanaka
Journal: Trans. Amer. Math. Soc. 361 (2009), 6205-6253
MSC (2000): Primary 35J60; Secondary 35J20
Published electronically: July 14, 2009
MathSciNet review: 2538593
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Abstract: We study the nonlinear Schrödinger equations:

$\displaystyle (P_\lambda)\quad\qquad\qquad-\Delta u+(\lambda^2 a(x)+1)u =\vert u\vert^{p-1}u, \quad u\in H^1(\mathbf{R}^N), \qquad\qquad\qquad$

where $ p>1$ is a subcritical exponent, $ a(x)$ is a continuous function satisfying $ a(x)\geq 0$, $ 0<\liminf_{\vert x\vert\to\infty} a(x)\leq \limsup_{\vert x\vert\to\infty}a(x)<\infty$ and $ a^{-1}(0)$ consists of 2 connected bounded smooth components $ \Omega_1$ and $ \Omega_2$.

We study the existence of solutions $ (u_\lambda)$ of $ (P_\lambda)$ which converge to 0 in $ \mathbf{R}^N\setminus(\Omega_1\cup\Omega_2)$ and to a prescribed pair $ (v_1(x),v_2(x))$ of solutions of the limit problem:

$\displaystyle -\Delta v_i+v_i=\vert v_i\vert^{p-1}v_i\quad \hbox{in} \Omega_i $

$ (i=1,2)$ as $ \lambda\to \infty$.


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Additional Information

Yohei Sato
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email: yohei-sato@aoni.waseda.jp

Kazunaga Tanaka
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email: kazunaga@waseda.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04565-6
PII: S 0002-9947(09)04565-6
Keywords: Nonlinear Schr\"odinger equations, singular perturbation, critical frequency, sign-changing solutions
Received by editor(s): June 8, 2005
Received by editor(s) in revised form: October 21, 2005, and May 10, 2007
Published electronically: July 14, 2009
Additional Notes: The second author was partially supported by Grant-in-Aid for Scientific Research (C) (2) (No. 17540205) and (B) (2) (No. 20340037) Japan Society for the Promotion of Science
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.