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) = λ F_u (x, u, v), in Ω,; - div (h₂ (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_loc¹ (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]. © 2009 Elsevier Inc. All rights reserved.

Chung N.T., Toan H.Q.

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