Differential and Integral Equations

An existence result for a class of superlinear $p$-Laplacian semipositone systems

D. D. Hai, C. Maya, and R. Shivaji

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In this paper we study positive solutions for the system \begin{align*} -(r^{N-1}\phi(u'))' & \, =\, \lambda r^{N-1}f(v);\ a\, <\,r\, <\,b; \\ -(r^{N-1}\phi(v'))' & \, =\, \lambda r^{N-1}g(u);\ a\, <\,r\, <\,b; \\ u(a)\, =\, 0 & \,=\,u(b)\,;\, v(a)\,=\,0\,=\,v(b), \end{align*} where $ \lambda > 0 $ is a parameter and $ \phi $ is an odd, increasing homeomorphism on $ \Bbb R $. Here $ f,\, g \in C[0,\infty) $ belong to a class of superlinear functions at $\infty$. In particular we allow $ f(0) $ or $ g(0) $ or both to be negative (semipositone system). We discuss the existence of a positive solution for $ \lambda $ small. Our proof is based on degree theory.

Article information

Differential Integral Equations, Volume 14, Number 2 (2001), 231-240.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B18: Positive solutions of nonlinear boundary value problems
Secondary: 35A05 35J65: Nonlinear boundary value problems for linear elliptic equations 35J70: Degenerate elliptic equations 47H11: Degree theory [See also 55M25, 58C30]


Hai, D. D.; Shivaji, R.; Maya, C. An existence result for a class of superlinear $p$-Laplacian semipositone systems. Differential Integral Equations 14 (2001), no. 2, 231--240. https://projecteuclid.org/euclid.die/1356123354

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