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 entail spectral bounds of the matrices associated with $f=(f_1,\ldots,f_m)$. We employ the degree theory from [A. Ćwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations 254 (2013), no. 3, 1120-1136] for tangent perturbations of maximal monotone operators in Banach spaces.