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

Abstract

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.