Differential and Integral Equations

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

Maya Chhetri, Shobha Oruganti, and R. Shivaji

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Consider the boundary-value problem \begin{equation} \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} \end{equation} 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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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