Positive solutions for classes of $p$-Laplacian equations

Abstract

We study positive $C^1(\bar{\Omega})$ solutions to classes of boundary value problems of the form \begin{eqnarray*} -\Delta_{p} u & = & g(\lambda,u)\mbox{ in } \Omega \\ u & = & 0 \mbox{ on } \partial \Omega, \end{eqnarray*} where $ \Delta_{p} $ denotes the p-Laplacian operator defined by $$ \Delta_{p} z:= \mbox{div}(|\nabla z|^{p-2}\nabla z);\, p > 1, $$ where $ \lambda > 0$ is a parameter and $ \Omega $ is a bounded domain in $ R^{N} $; $ N \geq 2 $ with $\partial \Omega$ of class $ C^{2}$ and connected. (If $N=1$, we assume that $\Omega$ is a bounded open interval.) In particular, we establish existence and multiplicity results for classes of nondecreasing, p-sublinear functions $g(\lambda,\cdot)$ belonging to $C^1([0,\infty))$. Our results also extend to classes of p-Laplacian systems. Our proofs are based on comparison methods.