In this paper, we study the existence of nontrivial solutions for the problem-△u=∫(x,u,v)+h1(x)inΩ -△v=g(x,u,v)h2(x)inΩ u=v=0 oneΩwhere Ω is bounded domain in RN and h1,h2 ∈L2 (Ω). The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities ∫ and g: s,|t|→+∞lim ∫(x,s,t)/s=s|,t|→+∞lim g(x,s,t)/t=λ+, unifrmly onΩ, -s,|t|→+∞lim ∫(x,s,t)/s=s|,t|→+∞lim g(x,s,t)/t=λ-, unifrmly onΩ,where λ+,λ-∈{0}∪σ(-△),σ(-△) denote the spectrum of-△. The cases (i) where λ+=λ- and (ii) where λ+=λ- such that the closed interval with endpoints λ+,λ- contains at most one simple eigenvalue of -△ are considered.
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