Electron. J. Diff. Equ., Vol. 2016 (2016), No. 59, pp. 1-7.

Infinitely many sign-changing solutions for Kirchhoff-type equations with power nonlinearity

Xianzhong Yao, Chunlai Mu

Abstract:
In this article we consider the Kirchhoff-type elliptic problem
$$\displaylines{
 -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^{p-2}u, \quad\text{in } \Omega,\cr
 u=0, \quad \text{on } \partial\Omega,
 }$$
where $\Omega\subset\mathbb{R}^N$ and $p\in(2,2^*)$ with $2^*=\frac{2N}{N-2}$ if $N\geq 3$, and $2^*=+\infty$ otherwise. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik-Schnirelman type minimax method.

Submitted December 3, 2015. Published February 29, 2016.
Math Subject Classifications: 35J60, 58E05, 34C14.
Key Words: Kirchhoff-type; sign-changing solutions; invariant sets of descent flow.

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Xianzhong Yao
College of Mathematics and Statistics
Chongqing University
Chongqing 401331, China
email: yaoxz416@163.com
Chunlai Mu
College of Mathematics and Statistics
Chongqing University
Chongqing 401331, China
email: clmu2005@163.com

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