Electron. J. Diff. Equ., Vol. 2013 (2013), No. 180, pp. 1-8.

Existence of positive solutions for Kirchhoff type equations

Ghasem A. Afrouzi, Nguyen Thanh Chung,Saleh Shakeri

Abstract:
In this article, we are interested in the existence of positive solutions for the Kirchhoff type problems
$$\displaylines{ -M\Big(\int_{\Omega}|\nabla u|^p\,dx\Big)\Delta_pu = \lambda f(u) \quad \hbox{in } \Omega,\cr u > 0 \quad \hbox{in } \Omega, \quad u =0 \quad \hbox{on } \partial\Omega, }$$
where $ 1<p< N $, $M : \mathbb{R}^+\to \mathbb{R}^+$ is a continuous and increasing function, $ \lambda $ is a parameter, $ f: [0,+\infty) \to \mathbb{R} $ is a $ C^1 $ nondecreasing function satisfying $ f(0)<0 $ (semipositone). Our proof is based on the sub- and super-solutions techniques.

Submitted April 23, 2013. Published August 07, 2013.
Math Subject Classifications: 35D05, 35J60.
Key Words: Kirchhoff type problems; semipositone; positive solution; sub-supersolution method.

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Ghasem Alizadeh Afrouzi
Department of Mathematics
Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: afrouzi@umz.ac.ir
Nguyen Thanh Chung
Dept. Science Management and International Cooperation
Quang Binh University, 312 Ly Thuong Kiet
Dong Hoi, Quang Binh, Vietnam
email: ntchung82@yahoo.com
Saleh Shakeri
Department of Mathematics
Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: s.shakeri@umz.ac.ir

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