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Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential. (English) Zbl 1100.35096
Summary: This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential $i\varphi_t=-\Delta\varphi+V(x)\varphi-\mu|\varphi|^{p-1} \varphi-\lambda|\varphi|^{q-1}\varphi,\;x\in \mathbb{R}^N,\;t\geq 0,$ where $$|mu>0$$, $$\lambda>0$$, and $$1<p<q<1+ 4/N$$. The potential $$V(x)$$ is bounded from below and satisfies $$V(x)\to \infty$$ as $$|x|\to\infty$$. From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.
##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 81V80 Quantum optics
##### Keywords:
Bose-Einstein condensate; compactness lemma; standing waves
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##### References:
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