TY - JOUR
T1 - Existence Results for a Class of Semilinear Elliptic Systems
JO - Journal of Partial Differential Equations
VL - 2
SP - 111
EP - 126
PY - 2009
DA - 2009/05
SN - 22
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5250.html
KW - Elliptic systems
KW - homotopy
KW - topological degree
AB - <p>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.</p>