Existence and multiplicity of nontrivial solutions for Klein-Gordon-Maxwell system with a parameter. (English) Zbl 1371.35106
Summary: This paper is concerned with the following Klein-Gordon-Maxwell system:
\[ \begin{cases} -\Delta u+ \lambda V(x)u-(2\omega+\phi)\phi u=f(x,u),\quad x\in \mathbb R^3,\\ \Delta\phi=(\omega+\phi)u^2,\quad x\in \mathbb R^3, \end{cases} \]
where \(\omega>0\) is a constant and \(\lambda\) is the parameter. Under some suitable assumptions on \(V(x)\) and \(f(x, u)\), we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
\[ \begin{cases} -\Delta u+ \lambda V(x)u-(2\omega+\phi)\phi u=f(x,u),\quad x\in \mathbb R^3,\\ \Delta\phi=(\omega+\phi)u^2,\quad x\in \mathbb R^3, \end{cases} \]
where \(\omega>0\) is a constant and \(\lambda\) is the parameter. Under some suitable assumptions on \(V(x)\) and \(f(x, u)\), we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
