Liu, Zeng; Huang, Yisheng Multiple solutions of asymptotically linear Schrödinger-Poisson system with radial potentials vanishing at infinity. (English) Zbl 1332.35112 J. Math. Anal. Appl. 411, No. 2, 693-706 (2014). 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)]. Cited in 6 Documents MSC: 35J50 Variational methods for elliptic systems 35J47 Second-order elliptic systems Keywords:Schrödinger-Poisson system; asymptotically linear; vanishing potentials Citations:Zbl 1136.35037 PDFBibTeX XMLCite \textit{Z. Liu} and \textit{Y. Huang}, J. Math. Anal. Appl. 411, No. 2, 693--706 (2014; Zbl 1332.35112) Full Text: DOI