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

Abstract

In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well \begin{equation*} \begin{cases} -\Delta u+(\lambda a(x)+1)u-\mu(2\omega+\phi)\phi u= f(x,u),& \text{in} \, \mathbb{R}^3,\\ \Delta \phi =\mu(\omega +\phi )u^2,& \text{in} \, \mathbb{R}^3, \end{cases} \end{equation*} where $\omega>0$ is a constant, $\mu$ and $\lambda$ are positive parameters, $f\in C(\mathbb{R}^3 \times \mathbb{R},\mathbb{R})$ and the nonlinearity $f$ satisfies the Ambrosetti–Rabinowitz condition. We use parameter-dependent compactness lemma to prove the existence of nontrivial solution for $\mu$ small and $\lambda$ large enough, then explore the asymptotic behavior as $\mu\rightarrow0$ and $\lambda\rightarrow\infty$. Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solution of Klein–Gordon–Maxwell system when $f(u):=|u|^{q-2}u$ where $2<q<4$.

Keywords

• klein–gordon–maxwell system
• asymptotic behavior
• variational method