On a singular elliptic system with quadratic growth in the gradient

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• 作者：Mohamed Benrhouma ; ^{mbenrhouma@uod.edu.sa}
• 关键词：Singular elliptic systems ; Quadratic growth in the gradient ; Sub-supersolutions ; Variational method ; Nehari manifold ; Concentration-compactness principle
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：448
• 期：2
• 页码：1120-1146
• 全文大小：491 K
• 卷排序：448

文摘

In this paper, we deal with the following singular elliptic system:{−Δu+α|∇u|2u=pp+qa(x)|v|q|u|p−2u+f,x∈RN,−Δv+β|∇v|2v=qp+qa(x)|u|p|v|q−2v+g,x∈RN, where N≥3N≥3, α,β>N+24, p,q>1 and p+q≤N+2N−2. We show through the sub- and supersolutions method, the existence of a nonnegative solution for an approximated system. The limit of the approximated solution is a positive solution. In the case, α=β=0α=β=0, p=qp=q and f=gf=g, we prove the uniqueness of a solution. Among others, we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle, a minimization method and with the help of Nehari manifold. The proofs rely on the concentration-compactness principle.