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Positive solutions for a nonlinear Schrödinger-Poisson system. (English) Zbl 1401.35042

Summary: In this paper, we study the following nonlinear Schrödinger-Poisson system \[ \begin{cases} -\Delta u + u + \varepsilon K(x)\Phi (x)u = f(u), \quad& x \in \mathbb{R}^3, \\ -\Delta \Phi = K (x)u^2, &x \in \mathbb{R}^3,\end{cases} \] where \(K(x)\) is a positive and continuous potential and \(f(u)\) is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some \(\varepsilon_0 > 0\) such that for \(0 <\varepsilon < \varepsilon_0\), the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result of W. Ao and J. Wei [Calc. Var. Partial Differ. Equ. 51, No. 3–4, 761–798 (2014; Zbl 1311.35077), Theorem 1.1] and a result of G. Li et al. [J. Math. Phys. 52, No. 5, 053505, 19 p. (2011; Zbl 1317.35238)].

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J47 Second-order elliptic systems
35B09 Positive solutions to PDEs
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