Electron. J. Diff. Equ., Vol. 2016 (2016), No. 70, pp. 1-13.

Elliptic systems at resonance for jumping non-linearities

Hakim Lakhal, Brahim Khodja

Abstract:
In this article, we study the existence of nontrivial solutions for the problem
$$\displaylines{ -\Delta u=\alpha _1u^{+}-\beta _1u^{-}+f(x,u,v)+h_1( x) \quad \text{in }\Omega, \cr -\Delta v=\alpha _2v^{+}-\beta _2v^{-}+g(x,u,v)+h_2( x) \quad \text{in }\Omega, \cr u=v=0\quad \text{on }\partial \Omega, }$$
where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$, and $h_1,h_2\in L^2( \Omega)$. Here $[ \alpha_j,\beta _j] \cap \sigma(-\Delta)={\lambda}$, where $\sigma(\cdot)$ is the spectrum. We use the Leray-Schauder degree theory.

Submitted July 26, 2015. Published March 15, 2016.
Math Subject Classifications: 35Q30, 65N12, 65N30, 76M25.
Key Words: Topological degree; elliptic systems; homotopy.

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Hakim Lakhal
Université de Skikda
B.P. 26 route d'El-Hadaiek, 21000, Algérie
email: H.lakhal@univ-skikda.dz
Brahim Khodja
Badji Mokhtar University
P.O. 12 Annaba, Algeria
email: brahim.khodja@univ-annaba.org

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