## 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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