J. Korean Math. Soc. 2020; 57(6): 1551-1571
Online first article September 10, 2020 Printed November 1, 2020
https://doi.org/10.4134/JKMS.j190833
Copyright © The Korean Mathematical Society.
Guofeng Che, Haibo Chen
Guangdong University of Technology; Central South University
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.
Keywords: Kirchhoff-Schr\"{o}dinger-Poisson system, concave and convex nonlinearities, Mountain Pass Theorem, Ekeland's variational principle
MSC numbers: 35J61, 35J20
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)
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