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Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. (English) Zbl 0857.35116

Consider the following nonlinear Schrödinger equation: \[ i\hslash\psi_t=- {\hslash^2\over2} \Delta\psi+V(x)\psi- \gamma|\psi |^{p-1}\psi.\tag{1} \] Using a Lyapunov-Schmidt type reduction with a standing wave solution
\(\psi(x,t)=\exp(-i \overline{E}t/\hslash) v(x)\), (1) is reduced to the equation on \(v(x)\): \[ -{\hslash^2\over2}\Delta v-(V(x)-\overline{E})v+ \gamma|\psi |^{p-1}v=0, \qquad x\in\mathbb{R}^N.\tag{2} \] The aim of this paper is to weaken the condition on \(V(x),\) \(\liminf_{x\to\infty} V(x)>\inf_{x\in\mathbb{R}^N} V(x)\) for the existence proof of (2) [cf. P. H. Rabinovitz, Z. Angew. Math. Phys. 43, No. 2, 270-291 (1992; Zbl 0763.35087)]. The author shows this condition can be replaced by \(\inf_{\partial\Omega} V(x)>\inf_\Omega V(x)\), for some bounded region \(\Omega\) in \(\mathbb{R}^N\).
Reviewer: A.Tsutsumi (Osaka)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs

Citations:

Zbl 0763.35087
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References:

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