Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells
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- by Yohei Sato and Kazunaga Tanaka PDF
- Trans. Amer. Math. Soc. 361 (2009), 6205-6253 Request permission
Abstract:
We study the nonlinear Schrödinger equations: \begin{equation*}(P_\lambda )\quad \qquad \qquad -\Delta u+(\lambda ^2 a(x)+1)u =|u|^{p-1}u, \quad u\in H^1(\mathbf {R}^N), \qquad \qquad \qquad \end{equation*} where $p>1$ is a subcritical exponent, $a(x)$ is a continuous function satisfying $a(x)\geq 0$, $0<\liminf _{|x|\to \infty } a(x)\leq \limsup _{|x|\to \infty }a(x)<\infty$ 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: \[ -\Delta v_i+v_i=|v_i|^{p-1}v_i\quad \mathrm {in}\; \Omega _i \] $(i=1,2)$ 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6205-6253
- MSC (2000): Primary 35J60; Secondary 35J20
- DOI: https://doi.org/10.1090/S0002-9947-09-04565-6
- MathSciNet review: 2538593