Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents

Abstract

In this paper, we study the existence and multiplicity of positive solutions to the following system − Δ u = ∂ F ∂ u ( u , v ) + ε g ( x ) , − Δ v = ∂ F ∂ v ( u , v ) + ε h ( x ) in Ω ; u , v > 0 in Ω ; and u = v = 0 on ∂ Ω , where Ω is a bounded smooth domain in R N ; F ∈ C 1 ( ( R + ) 2 , R + ) is positively homogeneous of degree μ ; g , h ∈ C 1 ( Ω ¯ ) ∖ { 0 } ; and ε is a positive parameter. Using sub–supersolution method, we prove the existence of positive solutions for the above problem. By means of the variational approach, we prove the multiplicity of positive solutions for the above problem with μ ∈ ( 2 , 2 ∗ ] .