Semi linear elliptic system at resonance

Document Type : Research Paper


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


In this work, we investigate the existence of weak solutions for the following semi-linear elliptic system
-\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 ,%
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.


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Volume 15, Issue 2
February 2024
Pages 369-377
  • Receive Date: 01 July 2023
  • Revise Date: 14 August 2023
  • Accept Date: 15 August 2023