Semi linear elliptic system at resonance

In this work, we investigate the existence of weak solutions for the following semi-linear elliptic system\begin{equation*}\left\{\begin{array}{c}-\Delta u+p(x)u=\alpha u+\phi \left( x,v\right) \ \ \ \ \text{in }\Omega ,\\-\Delta v+q(x)v=\beta v+\psi \left( x,u\right)  \ \ \ \  \text{in }\Omega ,%\end{array}\right.\end{equation*}with Dirichlet boundary condition, where \Omega is a bounded open set of \mathbb{R}^{N} \left( N\geq ۲\right) , \alpha ,\beta two real parameters, \left( p(x),q(x)\right) \in \left( L^{\infty }\left( \Omega \right) \right) ^{۲} and p(x),q(x)\geq ۰. using the Leray-Schauder's topological degree and under some suitable conditions for the non linearities \phi and \psi, we show the existence of nontrivial solutions.

کلیدواژه ها:

Homotopy ، boundary value problem ، fixed point theorems