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.
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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