Multiplicity of solutions for homogeneous elliptic systems with critical growth

• Published in: Journal of mathematical analysis and applications Vol. 385; no. 2; pp. 770 - 785
• Main Authors: Furtado, Marcelo F., da Silva, João Pablo P.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2012
• Subjects: Category of Ljusternik–Schnirelmann; Critical problems; Elliptic systems; Homogeneous problems; Critical problems; Elliptic systems; Homogeneous problems; Category of Ljusternik–Schnirelmann
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2011.07.001

In this paper we are concerned with the number of nonnegative solutions of the elliptic system { − Δ u = Q u ( u , v ) + 1 2 ⁎ H u ( u , v ) , in Ω , − Δ v = Q v ( u , v ) + 1 2 ⁎ H v ( u , v ) , in Ω , u = v = 0 , on ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, N ⩾ 4 , 2 ⁎ : = 2 N / ( N − 2 ) and Q u , H u and Q v , H v are the partial derivatives of the homogeneous functions Q , H ∈ C 1 ( R + 2 , R ) , where R + 2 : = [ 0 , ∞ ) × [ 0 , ∞ ) . In the proofs we apply variational methods and Ljusternik–Schnirelmann theory.