Well-posedness in weighted Sobolev spaces for elliptic equations of Cordes type

In this paper we prove some weighted W^2,2-a priori bounds for a class of linear, elliptic, second-order, differential operators of Cordes type in certain weighted Sobolev spaces on unbounded open sets Ω of R^n, n ≥ 2. More precisely, we assume that the leading coefficients of our differential operator satisfy the so-called Cordes type condition, which corresponds to uniform ellipticity if n = 2 and implies it if n ≥ 3, while the lower order terms are in specific Morrey type spaces. Here, our analytic technique mainly makes use of the existence of a topological isomorphism from our weighted Sobolev space, denoted by W^{2,2}_s (Ω) (s ∈ R), whose weight is a suitable function of class C^2( Ω), to the classical Sobolev space W^{2,2}(Ω), which allow us to exploit some well-known unweighted a priori estimates. Using the above mentioned W^{2,2}_s -a priori bounds, we also deduce some existence and uniqueness results for the related Dirichlet problems in the weighted framework.