Electron. J. Diff. Equ., Vol. 2016 (2016), No. 224, pp. 1-10.

Infinitely many solutions for Kirchhoff-type problems depending on a parameter

Juntao Sun, Yongbao Ji, Tsung-Fang Wu

Abstract:
In this article, we study a Kirchhoff type problem with a positive parameter $\lambda$,
$$\displaylines{ -K\Big( \int_{\Omega }|\nabla u|^{2}dx\Big) \Delta u=\lambda f(x,u) , \quad \text{in } \Omega , \cr u=0, \quad \text{on } \partial \Omega , }$$
where $K:[0,+\infty )\to \mathbb{R} $ is a continuous function and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a $L^{1}$-Caratheodory function. Under suitable assumptions on K(t) and f(x,u), we obtain the existence of infinitely many solutions depending on the real parameter $\lambda$. Unlike most other papers, we do not require any symmetric condition on the nonlinear term $f(x,u)$. Our proof is based on variational methods.

Submitted November 11, 2015. Published August 17, 2016.
Math Subject Classifications: 35B09, 35J20.
Key Words: Infinitely many solutions; Kirchhoff type problem; variational method.

An addendum was posted on September 29, 2016. It corrects Theorems 3.1 and 3.3. See the last three pages of this article.

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Juntao Sun
College of Science
Hohai University
Nanjing 210098, China
email: sunjuntao2008@163.com
Yongbao Ji
Institute of Finance and Economics
Shanghai University of Finance and Economics
Shanghai 200433, China
email: jiyongbao@126.com
Tsung-fang Wu
Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung 811, Taiwan
email: tfwu@nuk.edu.tw

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