We prove the existence of a positive solution of the Dirichlet problem for the Schrödinger equation whose potential possesses a critical singularity. It is assumed that the boundary of the domain is average concave in a neighborhood of the origin. Bibliography: 13 titles.

The problem of approximation of a given function on a given set by a polynomial of a fixed degree in the Chebyshev metric (the Chebyshev polynomial approximation problem) is a typical problem of Nonsmooth Analysis (to be more precise, it is a convex nonsmooth problem). It has many important applications, both in mathematics and in practice. The theory of Chebyshev approximations enjoys very nice properties (the most famous being the Chebyshev alternation rule).In the present article the problem of approximation of a given function on a given finite set of points by several polynomials is studied. As a criterion function, the maximin deviation is taken. The resulting functional is nonsmooth and nonconvex and therefore the problem becomes multiextremal and may have local minimizers which are not global ones.A necessary and sufficient condition for a point to be a local minimizer is proved. It is shown that a generalized alternation rule is still valid. Sufficient conditions for a point to be a strict local minimizer are established as well. These conditions are also formulated in terms of alternants. An exchange algorithm for finding a local minimizer is constructed. An "-exchange algorithm, allowing to find a ''better'' local minimizer is stated. Numerical examples illustrating the theory are presented.

Abstract. Sufficient conditions for the existence of extremal functions in Sobolevtype inequalities on manifolds with or without boundary are established. Some of these conditions are shown to be sharp. Similar results for embeddings in some weighted L q -spaces are obtained.

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