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Volume 22, Issue 2
Existence Results for a Class of Semilinear Elliptic Systems

Abdelkrim Moussaoui & Brahim Khodja

J. Part. Diff. Eq., 22 (2009), pp. 111-126.

Published online: 2009-05

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  • 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.

  • Keywords

Elliptic systems homotopy topological degree

  • AMS Subject Headings

Primary 35J55 15A15 Secondary 05A15 15A18

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-22-111, author = {}, title = {Existence Results for a Class of Semilinear Elliptic Systems}, journal = {Journal of Partial Differential Equations}, year = {2009}, volume = {22}, number = {2}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5250.html} }
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 -

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.

Abdelkrim Moussaoui & Brahim Khodja . (2019). Existence Results for a Class of Semilinear Elliptic Systems. Journal of Partial Differential Equations. 22 (2). 111-126. doi:
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