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Existence and nonexistence of the ground state solutions for nonlinear Schrödinger equations with nonperiodic nonlinearities. (English) Zbl 1256.35143

Summary: We study the following nonlinear Schrödinger equation \[ -\Delta u+ a(x)u= f(x,u),\qquad u\in H^1(\mathbb{R}^N), \] where \(a\) is periodic with respect to \(x\) and \(f(x,u)\) is superlinear in \(u\). Suppose that \(0\) lies in a gap of the spectrum \(\sigma(-\Delta+ a)\). Without periodic assumption on \(f\), we prove the existence of ground state solution for the system. Moreover, we obtain some sufficient conditions for the nonexistence of ground state solution.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
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