Differential and Integral Equations

Positive solutions for classes of $p$-Laplacian equations

Maya Chhetri, Shobha Oruganti, and R. Shivaji

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B50: Maximum principles 35J25: Boundary value problems for second-order elliptic equations


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

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