Electron. J. Diff. Equ., Vol. 2015 (2015), No. 280, pp. 1-12.

Existence of positive solutions for Kirchhoff problems

Jia-Feng Liao, Peng Zhang, Xing-Ping Wu

Abstract:
We study problems for the Kirchhoff equation
$$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =\nu u^3+ Q(x)u^{q},\quad \text{in }\Omega, \cr u=0, \quad \text{on }\partial\Omega, }$$
where $\Omega\subset \mathbb{R}^3$ is a bounded domain, $a,b\geq0$ and $a+b>0$, $\nu>0$, $3<q\leq5$ and $Q(x)>0$ in $\Omega$. By the mountain pass lemma, the existence of positive solutions is obtained. Particularly, we give a condition of Q to ensure the existence of solutions for the case of q=5.

Submitted June 25, 2015. Published November 10, 2015.
Math Subject Classifications: 35B09, 35B33, 35J20.
Key Words: Kirchhoff type equation; resonance; positive solution; mountain pass lemma.

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Jia-Feng Liao
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: liaojiafeng@163.com
Peng Zhang
School of Mathematics and Computational Science
Zunyi Normal College
Zunyi, China
email: gzzypd@sina.com
Xing-Ping Wu
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: wuxp@swu.edu.cn

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