Differential and Integral Equations

Existence results for a class of $p$-Laplacian problems with sign-changing weight

Abstract

Consider the boundary-value problem $$\begin{array}{c} -\Delta_p u = \lambda g(x)f(u) \quad \mbox{in}\ \Omega \nonumber\\ u=0 \quad \mbox{on}\quad \partial \Omega, \nonumber \end{array}$$ where $\lambda > 0$ is a parameter, $\Omega$ is a bounded domain in $\mathbb R^N,\, N \geq 1,$ with sufficiently smooth boundary $\partial \Omega$ and $\Delta_p u:= div(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator for $p > 1$. Here $g$ is a $C^1$ sign-changing function that may be negative near the boundary and $f$ is a $C^1$ nondecreasing function satisfying $f(0)>0$. We discuss existence results for positive solutions when $f$ satisfies certain additional conditions. We employ the method of sub-super solutions to obtain our results.

Article information

Source
Differential Integral Equations, Volume 18, Number 9 (2005), 991-996.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060118

Mathematical Reviews number (MathSciNet)
MR2162422

Zentralblatt MATH identifier
1212.35147

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations

Citation

Chhetri, Maya; Oruganti, Shobha; Shivaji, R. Existence results for a class of $p$-Laplacian problems with sign-changing weight. Differential Integral Equations 18 (2005), no. 9, 991--996. https://projecteuclid.org/euclid.die/1356060118