On a class of degenerate and singular elliptic systems in bounded domains

This paper deals with the nonexistence and multiplicity of nonnegative, nontrivial solutions to a class of degenerate and singular elliptic systems of the form{−div(h1(x)∇u)=λFu(x,u,v)in Ω,−div(h2(x)∇v)=λFv(x,u,v)in Ω, where Ω is a bounded domain with smooth boundary ∂Ω   in RNRN, N≧2N≧2, and hi:Ω→[0,∞), hi∈Lloc1(Ω), hihi (i=1,2i=1,2) are allowed to have “essential” zeroes at some points in Ω  , (Fu,Fv)=∇F(Fu,Fv)=∇F, and λ is a positive parameter. Our proofs rely essentially on the critical point theory tools combined with a variant of the Caffarelli–Kohn–Nirenberg inequality in [P. Caldiroli, R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl. 7 (2000) 189–199].