Dimitrios A. Kandilakis, Athanasios N. Lyberopoulos
Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\Bbb R^n$

Comment.Math.Univ.Carolinae 44,4 (2003) 645-658.

Abstract:We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$ \cases -div(|\nabla u|^{p-2}\nabla u) =f(x,u), \quad & x\in \Omega , u=0, & x\in \partial \Omega , \endcases $$ where $\Omega $is a bounded domain in $\Bbb R^n$, $1<p<+\infty $, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega )$ such that $0<u_{0}\leq u_{1}$ in $\Omega $, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.

Keywords: $p$-Laplacian, positive solutions, sub- and supersolutions, local minimizers, Palais-Smale condition
AMS Subject Classification: 35J20, 35J60, 35J70