Positive solutions to p-Laplace reaction-diffusion systems with nonpositive right-hand side
DOI:
https://doi.org/10.12775/TMNA.2015.065Słowa kluczowe
degree theory, tangency condition, cone, quasilinear elliptic system, positive weak solutionAbstrakt
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.
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