July  2003, 9(4): 1063-1071. doi: 10.3934/dcds.2003.9.1063

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, United States

2. 

Department of Mathematics and Statitstics, Mississippi State University, Mississippi State, MS 39762, United States, United States

Received  January 2002 Revised  October 2002 Published  April 2003

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. $

Citation: Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
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