## Topological Methods in Nonlinear Analysis

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

Mateusz Maciejewski

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 731-754.

Dates
First available in Project Euclid: 21 March 2016

https://projecteuclid.org/euclid.tmna/1458588659

Digital Object Identifier
doi:10.12775/TMNA.2015.065

Mathematical Reviews number (MathSciNet)
MR3494968

Zentralblatt MATH identifier
1381.35062

#### Citation

Maciejewski, Mateusz. Positive solutions to $p$-Laplace reaction-diffusion systems with nonpositive right-hand side. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 731--754. doi:10.12775/TMNA.2015.065. https://projecteuclid.org/euclid.tmna/1458588659

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