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Existence of solutions for Schrödinger-Poisson system with asymptotically periodic terms. (English) Zbl 1449.35211

Summary: In this paper, we consider the following nonlinear Schrödinger-Poisson system \[ \begin{cases} -\Delta u + V(x)u+K(x)\phi u= f(x,u), \quad & x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^{2}, & x\in \mathbb{R}^3, \end{cases} \] where \(V, K\in L^{\infty}(\mathbb{R}^3)\) and \(f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous. We prove that the problem has a nontrivial solution under asymptotically periodic case of \(V\), \(K\), and \(f\) at infinity. Moreover, the nonlinear term \(f\) does not satisfy any monotone condition.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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