Solutions of a Schrödinger-Kirchhoff-Poisson system with concave-convex nonlinearities. (English) Zbl 1526.35148
Summary: We consider the following nonlinear Schrödinger-Kirchhoff-Poisson system:
\[
\begin{cases}
\begin{aligned}
&\displaystyle -(a+b\int_{\mathbb{R}^3} |\nabla u|^2)\Delta u + \phi u=\mu g(x,u)+\lambda f(x,u), && x \in \mathbb{R}^3,\\
&\displaystyle -\Delta \phi =u^2, \qquad \lim_{\vert x \vert \rightarrow \infty} \phi (x)=0, &&x \in \mathbb{R}^3,
\end{aligned}
\end{cases}
\]
where \(a\), \(b> 0\). Under certain assumptions, we prove the existence of infinitely many solutions with high energy by using Fountain theorem.
MSC:
| 35J47 | Second-order elliptic systems |
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
Schrödinger-Kirchhoff-Poisson system; concave-convex nonlinearities; existence of infinitely many solutionsReferences:
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