Existence and multiplicity of solutions for Klein\u2013Gordon\u2013Maxwell systems with sign-changing potentials

Abstract

In this paper, we study the following nonlinear Klein–Gordon–Maxwell system: \t\t\t{−Δu+V(x)u−(2ω+ϕ)ϕu=f(x,u)+λh(x)|u|q−2u,x∈R3,Δϕ=(ω+ϕ)u2,x∈R3,(Pλ)\\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}$$ \\textstyle\\begin{cases} -\\Delta u+ V(x)u-(2\\omega +\\phi )\\phi u = f(x,u)+\\lambda h(x) \\vert u \\vert ^{q-2}u, & x\\in \\mathbb{R}^{3},\\\\ \\Delta \\phi = (\\omega +\\phi )u^{2}, & x\\in \\mathbb{R}^{3}, \\end{cases}\\displaystyle \\quad (\\mathrm{P_{\\lambda }}) $$\\end{document} where ω and λ are positive constants, V is a continuous function with negative infimum, qin (1,2), hin L^{frac{2}{2-q}}(mathbb{R}^{3}) is a positive potential function. Under the classic Ambrosetti–Rabinowitz condition, nontrivial solutions are obtained via the symmetric mountain pass theorem and the mountain pass theorem. In our paper, the nonlinearity F can also change sign and does not need to satisfy any 4-superlinear condition. We extend and improve some existing results to some extent.