## 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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