On the existence and multiplicity of solutions for nonlinear Klein–Gordon–Maxwell system

Wang, Lixia and Xiong, Chunlian and Zhao, Pingping (2023) On the existence and multiplicity of solutions for nonlinear Klein–Gordon–Maxwell system. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (19). pp. 1-18. ISSN 1417-3875

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Abstract

In this paper, we study the existence and multiplicity solutions for the following Klein–Gordon–Maxwell system −∆u+V(x)u−(2ω+ϕ)ϕu = f(x,u), x ∈ R3, ∆ϕ =(ω+ϕ)u2, x ∈R3, where ω > 0is a constant and the nonlinearity f(x,u) is either asymptotically linear in u at infinity or the primitive of f(x,u) is of 4-superlinear growth in u at infinity. Under some suitable assumptions, the existence and multiplicity of solutions are proved by using the Mountain Pass theorem and the fountain theorem, respectively.

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