Abstract:

In this paper, we will study the existence of ground state solutions for a class of nonlinear equations by using the theory of compactness of concentration, variational method and critical point theory.

$\begin{eqnarray*} \left \{ \begin{array}{l} -\Delta u+(m+2\omega\phi)u=A(x)|u|^{p-2}u,\\ -\Delta\phi+\lambda\phi=\omega u^{2}, \lim\limits_{|x|\rightarrow\infty}u(x)=0, \lim\limits_{|x|\rightarrow\infty}\phi(x)=0. \end{array} \right. \end{eqnarray*}$

where $u\in H^{1}({\Bbb R}^{3})$, $\phi\in H^{1}({\Bbb R}^{3})$, $\lambda>0$, $m$ and $\omega$ are positive constants. Then we study the problem assuming the follwwing two cases on $A(x)$.

If $A(x)$ is a positive constant function, we prove that the ground state solution $(u, \phi)$ exists for any $p\in(4,6)$; if $A(x)$ is not a constant function, we prove that the ground state solution $(u, \phi)$ exists for any $p\in(4,6)$ under the right conditions.

Key words: Klein-Gordon-Maxwell equation, Principle of concentration compactness, Variational methods, Critical point theory, Ground state solution