Existence and uniqueness results for Navier problems with degenerated operators

Abstract

Ω⊂RN is a bounded open set, f ω2 ∈Lp (Ω, ω2), G ν2 ∈ [Ls (Ω, ν2)] , ω1, ω2, ν1 and ν2 are four weight functions (i.e., ωi and νi, i = 1, 2 are locally integrable functions on RN such that 0 < ωi(x), νi(x) < ∞ a.e. x∈RN), ∆ is the Laplacian operator, 1 < q, s < p < ∞, 1/p + 1/p ′ = 1 and 1/s + 1/s ′ = 1. For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1–8]). The type of a weight depends on the equation type. A class of weights, which is particularly well understood, is the class of Ap weights that was introduced by B.Muckenhoupt in the early 1970’s (see [7]). These classes have found many useful applications in harmonic analysis (see [9] and [10]). Another reason for studying Ap-weights is the fact that powers of the distance to submanifolds of RN often belong to Ap (see [8] and [11]). There are, in fact, many interesting examples of weights (see [6] for p-admissible weights). In the non-degenerate case (i.e. with ω(x) ≡ 1), for all f ∈ Lp(Ω) the Poisson equation associated with the Dirichlet problem { −∆u = f (x), in Ω u(x) = 0, in ∂Ω