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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 \(\varOmega \) of \(\mathbb {R}^{n},\,n\ge 2\). 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 \(n\ge 3\), 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 \(W^{2,2}_s(\varOmega )\) (\(s\in \small \mathbb {R}\)), whose weight is a suitable function of class \(C^2(\bar{\varOmega })\), to the classical Sobolev space \(W^{2,2}(\varOmega )\), 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.

We give an overview on some results concerning the unique solvability of the Dirichlet problem inW2,p,p>1, for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.

In this paper we prove some a priori bounds for the solutions of the Dirichlet problem for elliptic equations with singular coefficients in weighted Sobolev spaces.

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝn, n ≥ 2, not necessarily bounded or regular.

We prove an existence and uniqueness result for the Dirichlet problem for a class of elliptic equations with singular data in weighted Sobolev spaces.

In this paper we study certain weighted Sobolev spaces defined on an open subset @W of R^n (not necessarily bounded or regular) when the weight is a function related to the distance from a subset of @?@W. As an application, we prove boundedness and compactness results for operators in such weighted Sobolev spaces.

We deepen the study of some Morrey type spaces, denoted byMp,λ(Ω), defined on an unbounded open subsetΩofℝn. In particular, we construct decompositions for functions belonging to two different subspaces ofMp,λ(Ω), which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the above-mentioned subspaces.

In this paper we prove some interpolation inequalities between functions and their derivatives in the class of weighted Sobolev spaces defined on unbounded open subset Ω ⊂ R n .