Positive solutions to p-Laplace reaction-diffusion systems with nonpositive right-hand side

Abstract

The aim of the paper is to show the existence of positive solutions to the elliptic system of partial differential equations involving the $p$-Laplace operator\[\begin{cases}-\Delta_p u_i(x) = f_i(u_1 (x),u_2(x),\ldots,u_m(x)), & x\in \Omega,\ 1\leq i\leq m,\\u_i(x)\geq 0, & x\in \Omega,\ 1\leq i\leq m,\\u(x) = 0, & x\in \partial \Omega.\end{cases}\]We consider the case of nonpositive right-hand side $f_i$, $i=1,\ldots,m$. The sufficient conditions entails spectral bounds of the matrices associated with $f=(f_1,\ldots,f_m)$. We employ the degree theory from \cite{CwMac} for tangent perturbations of maximal monotone operators in Banach spaces.