Document Type : Research Paper

Author

• Ouahiba Gharbi

Departement of Mathematics, Faculty of Science; Badji Mokhtar University, Annaba, Algeria

Abstract

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 $\mathbb{R}^{N}$ $\left( N\geq 2\right) ,$ $\alpha ,\beta $ two real parameters, $\left( p(x),q(x)\right) \in \left( L^{\infty }\left( \Omega \right) \right) ^{2}$ and $p(x),q(x)\geq 0.$ 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.

Keywords

• Homotopy
• Boundary value problem
• Fixed point theorems