On positive solutions for classes of p-Laplacian semipositone systems

Maya Chhetri; D. D. Hai; R. Shivaji

doi:10.3934/dcds.2003.9.1063

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2003, Volume 9, Issue 4: 1063-1071. Doi: 10.3934/dcds.2003.9.1063

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On positive solutions for classes of p-Laplacian semipositone systems

1.
Department of Mathematical Sciences, The University of North Carolina at Greensboro, Greensboro, NC 27402
2.
Department of Mathematics and Statitstics, Mississippi State University, Mississippi State, MS 39762

Received: January 2002

Revised: October 2002

Published: July 2003

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Abstract

We study positive solutions for the system

$-\Delta_p u = \lambda f(v)$ in $\quad \Omega $

$-\Delta_p v = \lambda g(u)$ in $ \quad \Omega $

$u = 0 = v$ on $ \quad \partial \Omega$

where $ \lambda > 0 $ is a parameter, $ \Delta_p $ denotes the p-Laplacian operator defined by $ \Delta_p(z)$:=div$(|\nabla z|^{p-2}\nabla z) $ for $ p> 1 $ and $ \Omega $ is a bounded domain with smooth boundary. Here $ f,g \in C[0,\infty) $ belong to a class of functions satisfying $ \lim_{z \to \infty}\frac{f(z)}{z^{p-1}}=0, \lim_{z \to \infty}\frac{g(z)}{z^{p-1}}=0 $. In particular, we discuss the existence of radial solutions for large $ \lambda $ when $ \Omega $ is an annulus. For a general bounded region $ \Omega, $ we also discuss a non-existence result when $ f(0) < 0 $ and $ g(0) < 0. $

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Mathematics Subject Classification: Primary: 34B18, 35J55.

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