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Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations. (English) Zbl 1142.35601

Summary: In this paper we are concerned with multi-lump bound states of the nonlinear Schrödinger equation \[ i\hbar \frac{\partial\psi}{\partial t} = -\hbar^2 \Delta \psi + V\psi - \gamma |\psi|^{p-2}\psi \] for sufficiently small \(\hbar>0\), where \(\gamma>0\), \(2<p<\frac{2N}{N-2}\) for \(N\geq3\) and \(2<p<+\infty\) for \(N=1,2\). \(V\) is bounded on \(\mathbb R^N\). For any finite collection \(\{a^1,\ldots,a^k\}\) of nondegenerate critical points of \(V\), we show the uniqueness of solutions of the form \(e^{-iEt/\hbar}u(x)\) for \(E<\inf_{x \in\mathbb R^N} V(x)\), where \(u\) is positive on \(\mathbb R^N\) and is a small perturbation of a sum of one-lump solutions concentrated near \(a^1,\ldots,a^k\), respectively for sufficiently small \(\hbar>0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J35 Variational methods for higher-order elliptic equations
35J60 Nonlinear elliptic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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