Differential and Integral Equations

Multiple positive solutions for classes of $p$-Laplacian equations

Mythily Ramaswamy and Ratnasingham 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 & = & \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

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

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


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

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