Electron. J. Diff. Eqns., Vol. 2008(2008), No. 61, pp. 1-12.

Three solutions for singular p-Laplacian type equations

Zhou Yang, Di Geng, Huiwen Yan

Abstract:
In this paper, we consider the singular $p$-Laplacian type equation
$$\displaylines{ -\hbox{div}(|x|^{-\beta} a(x,\nabla u)) =\lambda f(x,u),\quad \hbox{in }\Omega,\cr u=0,\quad \hbox{on }\partial\Omega, }$$
where $0\leq\beta<N-p$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ containing the origin, $f$ satisfies some growth and singularity conditions. Under some mild assumptions on $a$, applying the three critical points theorem developed by Bonanno, we establish the existence of at least three distinct weak solutions to the above problem if $f$ admits some hypotheses on the behavior at $u=0$ or perturbation property.

Submitted January 14, 2008. Published April 22, 2008.
Math Subject Classifications: 35J60.
Key Words: p-Laplacian operator; singularity; multiple solutions.

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Zhou Yang
School of Math. Sci., South China Normal University
Guangzhou 510631, China
email: yangzhou@scnu.edu.cn
Di Geng
School of Math. Sci., South China Normal University
Guangzhou 510631, China
email: gengdi@scnu.edu.cn
Huiwen Yan
School of Math. Sci., South China Normal University
Guangzhou 510631, China
email: hwyan10@yahoo.com.cn

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