## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 6 (2003), 757-768.

Dates
First available in Project Euclid: 21 December 2012

Chhetri, Maya; Oruganti, Shobha; Shivaji, R. Positive solutions for classes of $p$-Laplacian equations. Differential Integral Equations 16 (2003), no. 6, 757--768. https://projecteuclid.org/euclid.die/1356060611