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
$$\{\begin{array}{c}-\mathrm{\Delta }u+p(x)u=\alpha u+\varphi (x,v)\text{in }\mathrm{\Omega },\\ -\mathrm{\Delta }v+q(x)v=\beta v+\psi (x,u)\text{in }\mathrm{\Omega },\end{array}$$
with Dirichlet boundary condition, where $\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^N$ $(N\ge 2),$ $\alpha ,\beta$ two real parameters, $(p(x),q(x))\in {\left(L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\right)}^2$ and $p(x),q(x)\ge 0.$ using the Leray-Schauder's topological degree and under some suitable conditions for the non linearities $\varphi$ and $\psi$, we show the existence of nontrivial solutions.

Keywords

• Homotopy
• Boundary value problem
• Fixed point theorems