Year: 2009
Journal of Partial Differential Equations, Vol. 22 (2009), Iss. 2 : pp. 111–126
Abstract
In this paper, we study the existence of nontrivial solutions for the problem -Δu=f(x,u,v)+h_1(x) in Ω, -Δv=g(x,u,v)+h_2(x) in Ω, u=v=0 on ∂Ω, where Ω is bounded domain in R^N and h_1,h_2∈L^2(Ω). The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g: \lim_{s,|t|→+∞}\frac{f(x,x,t)}{s}=\lim_{|s|,t→+∞}\frac{g(x,s,t)}{t}=λ_+, uniformly on Ω, \lim_{-s,|t|→+∞}\frac{f(x,x,t)}{s}=\lim_{|s|,-t→+∞}\frac{g(x,s,t)}{t}=λ_-, uniformly on Ω, where λ_+, λ_-∉{0}∪ σ(-Δ), σ(-Δ) denote the spectrum of -Δ. 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 -Δ are considered.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2009-JPDE-5250
Journal of Partial Differential Equations, Vol. 22 (2009), Iss. 2 : pp. 111–126
Published online: 2009-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Elliptic systems