Abstract
We consider existence and qualitative properties of standing wave solutions \(\Psi(x,t) = e^{-iEt/h}u(x)\) to the nonlinear Schrödinger equation \(ih\frac{\partial\psi} {\partial t} = -\frac{h^2}{2m}\Delta\psi+W(x)\psi-|\psi|^{p-1}\psi = 0\) with E being a critical frequency in the sense that inf \(_{x\in\mathbb{R}^N}W(x)=E\). We verify that if the zero set of W − E has several isolated points x i (\(i=1,\ldots,m\)) near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, \(1\leq k\leq m\), a standing wave solution which concentrates simultaneously on \(\{x_j|j=1,\ldots,k\}\), where \(\{x_j|j=1,\ldots,k\}\) is any given subset of \(\{x_i|i=1,\ldots,m\}\). This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).
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Supported by the Fund for Distinguished Young Scholars of NSFC in China.
Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).
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Cao, D., Peng, S. Multi-bump Bound States of Schrödinger Equations with a Critical Frequency. Math. Ann. 336, 925–948 (2006). https://doi.org/10.1007/s00208-006-0021-y
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DOI: https://doi.org/10.1007/s00208-006-0021-y