Electron. J. Diff. Equ., Vol. 2014 (2014), No. 10, pp. 1-10.

Solutions to Kirchhoff equations with combined nonlinearities

Ling Ding, Lin Li, Jing-Ling Zhang

Abstract:
We prove the existence of multiple positive solutions for the Kirchhoff equation
$$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =h(x)u^q+f(x,u), \quad x\in \Omega, \cr u=0, \quad x\in\partial \Omega, }$$
Here $\Omega $ is an open bounded domain in $ R^{N}$ ( $N=1,2,3$), $h(x)\in L^\infty(\Omega)$, $f(x,s)$ is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. Our main tools are the Ekeland's variational principle and the mountain pass lemma.

Submitted July 22, 2013. Published January 7, 2014.
Math Subject Classifications: 35J60, 35J40, 35B38.
Key Words: Kirchhoff equation; asymptotically linear; asymptotically 3-linear; positive solution; mountain pass lemma.

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Ling Ding
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: dingling1975@qq.com
Lin Li
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email:lilin420@gmail.com
Jing-Ling Zhang
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: 1293503066@qq.com

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