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Positive solutions for some non-autonomous Schrödinger-Poisson systems. (English) Zbl 1183.35109

Summary: We study the Schrödinger-Poisson system
\[ \begin{cases} -\Delta u+u+K(x) \varphi(x)u= a(x)|u|^{p-1}u, &x\in\mathbb R^3,\\ -\Delta\varphi= K(x)u^2, &x\in\mathbb R^3, \end{cases} \]
with \(p\in(3,5)\). Assuming that \(a:\mathbb R^3\to\mathbb R\) and \(K:\mathbb R^3\to\mathbb R\) are nonnegative functions such that
\[ \lim_{|x|\to\infty} a(x)= a_\infty>0, \qquad \lim_{|x|\to\infty} K(x)=0 \]
and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive solutions.

MSC:

35J47 Second-order elliptic systems
35B09 Positive solutions to PDEs
35D30 Weak solutions to PDEs
35J50 Variational methods for elliptic systems
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