Standing wave concentrating on compact manifolds for nonlinearSchrödinger equations

Jaeyoung Byeon; Ohsang Kwon; Yoshihito Oshita

doi:10.3934/cpaa.2015.14.825

Article Contents

2015, Volume 14, Issue 3: 825-842. Doi: 10.3934/cpaa.2015.14.825

This issue Previous Article Uniform stability of the Boltzmann equation with an external force near vacuum Next Article Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

1.
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu Daejeon, 305-701
2.
Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai
3.
Department of Mathematics, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530

Received: March 31, 2014

Revised: November 30, 2014

Published: April 2015

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Abstract

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.

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Mathematics Subject Classification: 35J20, 35Q55, 35B40.

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