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On the Schrödinger-Poisson system with indefinite potential and 3-sublinear nonlinearity. (English) Zbl 1436.35082
Summary: We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity \(f(u)\) and coercive sign-changing potential \(V\) so that the Schrödinger operator \(-\Delta + V\) is indefinite. Previous results in this framework required \(f\) to be strictly 3-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where \(f(t) = |t|^2 t\); in this paper we fill this gap, obtaining non-trivial solutions when \(f\) is not necessarily 3-superlinear.

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
35J47 Second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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