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Multiple solutions of asymptotically linear Schrödinger-Poisson system with radial potentials vanishing at infinity. (English) Zbl 1332.35112

Summary: In this paper we consider the following Schrödinger-Poisson system \[ \begin{cases} -\Delta u+V(x)u+\lambda\phi u=Q(x)f(u),\quad & x\in\mathbb R^3,\\-\Delta\phi=u^2,\quad & x\in\mathbb R^3,\end{cases} \] where potentials \(V\sim|x|^{-\alpha}\), \(Q\sim|x|^{-\beta}\) \((\alpha>0,\beta>0\), and \(f:\mathbb R\to\mathbb R\) is asymptotically linear at infinity. Working in a variational setting, we prove the existence and multiplicity of solutions for the system when \(\lambda\) is small and \(\alpha,\beta\) belong to different ranges. We also study the boundedness of the set of the solutions found, as \(\lambda\to 0^+\), in the spirit of D. Ruiz [J. Funct. Anal. 237, No. 2, 655–674 (2006; Zbl 1136.35037)].

MSC:

35J50 Variational methods for elliptic systems
35J47 Second-order elliptic systems

Citations:

Zbl 1136.35037
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