Existence and multiplicity of nontrivial solutions for Schr\xf6dinger-Poisson systems on bounded domains

Abstract

In this paper, we concern with the following Schrödinger-Poisson system: {−Δu+ϕu=f(x,u),x∈Ω,−Δϕ=u2,x∈Ω,u=ϕ=0,x∈∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u+\\phi u = f(x,u) , & x\\in\\Omega,\\\\ -\\Delta\\phi=u^{2}, & x\\in\\Omega,\\\\ u=\\phi=0, & x \\in\\partial\\Omega, \\end{cases} $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{3}. Under more appropriate assumptions on f, we obtain new results on the existence of nontrivial solutions and infinitely many solutions by using the mountain pass theorem and the symmetric mountain pass theorem, respectively. We extend and improve some recent results in the literature.