Multi-bump solutions for  Schrödinger equation involving critical growth and  potential wells

Yuxia Guo; Zhongwei Tang

doi:10.3934/dcds.2015.35.3393

Article Contents

2015, Volume 35, Issue 8: 3393-3415. Doi: 10.3934/dcds.2015.35.3393

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Multi-bump solutions for Schrödinger equation involving critical growth and potential wells

Yuxia Guo^1, and
Zhongwei Tang^2,

1.
Department of Mathematics, Tsinghua University, Beijing, 100084
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: September 30, 2014

Revised: November 30, 2014

Published: May 2017

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Abstract

In this paper, we consider the following Schrödinger equation with critical growth $$-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u \quad \hbox{ in } \mathbb{R}^N, $$ where $N\geq 5$, $2^*$ is the critical Sobolev exponent, $\delta>0$ is a constant, $a(x)\geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components $\Omega_1,\cdots,\Omega_k$ such that the interior of $\Omega_i (i=1, 2, ..., k)$ is not empty and $\partial\Omega_i (i=1, 2, ..., k)$ is smooth, then for any non-empty subset $J\subset \{1,2,\cdots,k\}$ and $\lambda$ sufficiently large, the equation admits a solution which is trapped in a neighborhood of $\bigcup_{j\in J}\Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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