Existence and Multiplicity Results for a Degenerate Elliptic Equation

Abstract

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub– and supersolutions argument for p–Laplacian operators, under suitable conditions on the nonlinearity f(x, s), we show the following problem: $$ \begin{array}{*{20}c} {{ - \Delta _{p} u = \lambda u^{\alpha } - a{\left( x \right)}u^{q} \;{\text{in}}\;\Omega ,}} & {{\left. u \right|_{{\partial \Omega }} }} \\ \end{array} = 0 $$ possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R N , N ≥ 2, with C 2 boundary, λ is a positive parameter, Δ p is the p–Laplacian operator with p > 1, α, q are given constants such that p − 1 < α < q, and a(x) is a continuous positive function in $$ \ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega } $$ Ω.