Existence and multiplicity of solutions for Kirchhoff- Schr\"{o}dinger-Poisson system with concave and convex nonlinearities

J. Korean Math. Soc. Published online September 10, 2020

Guofeng Che and Haibo Chen
Guangdong University of Technology, Central South University

Abstract : This paper is concerned with the following Kirchhoff-Schr\"{o}dinger-Poisson system
$$
\left\{
\begin{array}{ll}
\displaystyle
-\big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u
+ V(x)u+\mu\phi u=\lambda f(x)|u|^{p-2}u+g(x)|u|^{q-2}u, &\mbox{ \ \ in }\mathbb{R}^{3},\\
-\Delta \phi= \mu |u|^{2}, &\mbox{ \ \ in }\mathbb{R}^{3},\\
\end{array}
\right.
$$
where $a>0,~b,~\mu\geq0$, $p\in(1,2)$, $q\in[4,6)$ and $\lambda>0$ is a parameter.
Under some suitable assumptions on $V(x)$, $f(x)$ and $g(x)$, we prove that
the above system has at least two different nontrivial solutions via the
Ekeland's variational principle and the Mountain Pass Theorem in critical
point theory. Some recent results from the literature are improved and extended.