Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR KIRCHHOFF-SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES

  • Che, Guofeng (School of Applied Mathematics Guangdong University of Technology) ;
  • Chen, Haibo (School of Mathematics and Statistics Central South University)
  • Received : 2019.12.11
  • Accepted : 2020.04.24
  • Published : 2020.11.01
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Abstract

This paper is concerned with the following Kirchhoff-Schrödinger-Poisson system $$\begin{cases} -(a+b{\displaystyle\smashmargin{2}\int\nolimits_{\mathbb{R}^3}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+{\mu}{\phi}u={\lambda}f(x){\mid}u{\mid}^{p-2}u+g(x){\mid}u{\mid}^{p-2}u,&{\text{ in }}{\mathbb{R}}^3,\\-{\Delta}{\phi}={\mu}{\mid}u{\mid}^2,&{\text{ in }}{\mathbb{R}}^3, \end{cases}$$ where a > 0, b, µ ≥ 0, p ∈ (1, 2), q ∈ [4, 6) and λ > 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.

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References

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