Electron. J. Diff. Equ., Vol. 2015 (2015), No. 215, pp. 1-7.

Multiple solutions for Kirchhoff type problem near resonance

Shu-Zhi Song, Chun-Lei Tang, Shang-Jie Chen

Abstract:
Based on Ekeland's variational principle and the mountain pass theorem, we show the existence of three solutions to the Kirchhoff type problem
$$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx \Big) \Delta u =b \mu u^3+f(x,u)+h(x), \quad\text{in } \Omega, \cr u=0, \quad \text{on } \partial \Omega. }$$
Where the parameter $\mu$ is sufficiently close, from the left, to the first nonlinear eigenvalue.

Submitted March 23, 2015. Published August 17, 2015.
Math Subject Classifications: 35J61, 35C06, 35J20.
Key Words: Near resonance; mountain pass theorem; Kirchhoff type; Ekeland's variational principle.

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Shu-Zhi Song
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: sjrdj@163.com
Chun-Lei Tang
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: tangcl@swu.edu.cn, Tel +8613883159865
Shang-Jie Chen
School of Mathematics and Statistics
Chongqing Technology and Business University
Chongqing 400067, China
email: chensj@ctbu.edu.cn, 11183356@qq.com

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