Existence and asymptotic behavior of nontrivial solution for Klein–Gordon–Maxwell system with steep potential well

Wen, Xueping and Chen, Chunfang (2023) Existence and asymptotic behavior of nontrivial solution for Klein–Gordon–Maxwell system with steep potential well. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (17). pp. 1-18. ISSN 1417-3875

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Abstract

In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well −∆u+(λa(x)+1)u−µ(2ω+ϕ)ϕu = f(x,u), inR3, ∆ϕ =µ(ω+ϕ)u2, inR3, where ω > 0 is a constant, µ and λ are positive parameters, f ∈ C(R3 ×R,R) and the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition. We use parameterdependent compactness lemma to prove the existence of nontrivial solution for µ small and λ large enough, then explore the asymptotic behavior as µ → 0 and λ → ∞. Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solutions of the Klein–Gordon–Maxwell system when f(u) := |u|q−2u where 2 <q<4.

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