## Differential and Integral Equations

### Multiple 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 & = & \lambda f(u)\mbox{ in } \Omega \\ u & = & 0 \mbox{ on } \partial \Omega , \end{eqnarray*} where $\Delta_{p}$ denotes the p-Laplacian operator defined by $$\Delta_p z:= \mbox{div}(|\nabla z|^{p-2}\nabla z);\, p > 1,\, \lambda > 0$$ 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 of three positive solutions for classes of nondecreasing, p-sublinear functions $f$ belonging to $C^1([0,\infty))$. Our proofs are based on sub-supersolution techniques.

#### Article information

Source
Differential Integral Equations Volume 17, Number 11-12 (2004), 1255-1261.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060244

Mathematical Reviews number (MathSciNet)
MR2100025

Zentralblatt MATH identifier
1150.35419

Subjects
Primary: 35J60: Nonlinear elliptic equations

#### Citation

Ramaswamy, Mythily; Shivaji, Ratnasingham. Multiple positive solutions for classes of $p$-Laplacian equations. Differential Integral Equations 17 (2004), no. 11-12, 1255--1261.https://projecteuclid.org/euclid.die/1356060244