Find in Library

In this paper, we consider the singular $p$-Laplacian type equation $$displaylines{ -hbox{div}(|x|^{-eta} a(x, abla u)) =lambda f(x,u),quad hbox{in }Omega,cr u=0,quad hbox{on }partialOmega, }$$ where $0leqeta<N-p$, $Omega$ 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.