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

Abstract

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 ( h 1 ( x ) ∇ u ) = λ F u ( x , u , v ) in Ω , − div ( h 2 ( x ) ∇ v ) = λ F v ( x , u , v ) in Ω , where Ω is a bounded domain with smooth boundary ∂ Ω in R N , N ≧ 2 , and h i : Ω → [ 0 , ∞ ) , h i ∈ L loc 1 ( Ω ) , h i ( i = 1 , 2 ) are allowed to have “essential” zeroes at some points in Ω , ( F u , F v ) = ∇ 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].