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

Mateusz Maciejewski

DOI: http://dx.doi.org/10.12775/TMNA.2015.065


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
-\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.
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.


degree theory; tangency condition; cone; quasilinear elliptic system; positive weak solution

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