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

J. Korean Math. Soc. 2020 Vol. 57, No. 6, 1551-1571 https://doi.org/10.4134/JKMS.j190833 Published online September 10, 2020 Printed November 1, 2020

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

Abstract : This paper is concerned with the following Kirchhoff-Schr\"{o}d\-inger-Poisson system $$ \scalebox{0.84}{$\displaystyle\left\{\!\! \begin{array}{ll} \displaystyle -\big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi u=\lambda f(x)|u|^{p-2}u+g(x)|u|^{q-2}u, &\mbox{ in }\mathbb{R}^{3},\\ -\Delta \phi= \mu |u|^{2}, &\mbox{ in }\mathbb{R}^{3},\\ \end{array} \right.$} $$ 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.

Supported by : The first author was supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110275). The second author was supported by National Natural Science Foundation of China (Grant No. 11671403)