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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
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References:
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