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Standing waves with large frequency for 4-superlinear Schrödinger-Poisson systems. (English) Zbl 1309.35008
Summary: We consider standing waves with frequency $$\omega$$ for 4-superlinear Schrödinger-Poisson systems. For large $$\omega$$, the problem reduces to a system of elliptic equations in $$\mathbb R ^3$$ with potential indefinite in sign. The variational functional does not satisfy the mountain pass geometry. The nonlinearity considered here satisfies a condition which is much weaker than the classical (AR) condition and the condition (Je) of Jeanjean. We obtain nontrivial solutions and, in case of odd nonlinearity, an unbounded sequence of solutions via the local linking theorem and the fountain theorem, respectively.

MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J60 Nonlinear elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35J50 Variational methods for elliptic systems 35J47 Second-order elliptic systems
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