Multiple solutions for nonhomogeneous Schrodinger-Poisson system with p-Laplacian

Abstract

This article concerns the existence of solutions to the Schrodinger-Poisson system $$\displaylines{ -\Delta_p u+|u|^{p-2}u+\lambda\phi u=|u|^{q-2}u+h(x) \quad \hbox{in }\mathbb{R}^3,\\ -\Delta \phi=u^2 \quad \hbox{in }\mathbb{R}^3, }$$ where \( 4/3 < p < 12/5 \), \( p < q < p^{*}=3p/(3-p) \), \(\Delta_p u =\hbox{div}(|\nabla u|^{p-2}\nabla u)\), \(\lambda >0\), and \(h \not= 0\). The multiplicity results are obtained by using Ekeland's variational principle and the mountain pass theorem.