Sobolev-type regularity and Pohozaev-type identities for some degenerate and singular problems

Abstract

on the bottom of a half (N + 1)-dimensional ball. The interest in such a type of equations and related regularity issues has developed starting from the pioneering paper [7], proving local Hölder continuity results and Harnack’s inequalities, and has grown significantly in recent years stimulated by the study of the fractional Laplacian in its realization as a Dirichlet-to-Neumann map [3]. In this context, among recent regularity results for problems of type (1)–(2), we mention [2] and [12] for Schauder and gradient estimates with A being the identity matrix and c ≡ 0. More general degenerate/singular equations of type (1), admitting a varying coefficient matrix A, are considered in [19, 20]. In [19], under suitable regularity assumptions on A and c, Hölder continuity and C-regularity are established for solutions to (1)–(2) in the case h ≡ g ≡ 0, which, up to a reflection through the hyperspace t = 0, corresponds to the study of solutions to the equation − div(|t|1−2sA∇U) + |t|1−2sc = 0 which are even with respect to the t-variable; Hölder continuity of solutions which are odd in t is instead investigated in [20]. In addition, in [19] C and C bounds are derived for some inhomogeneous Neumann boundary problems (i.e. for g 6≡ 0) in the case c ≡ 0. The goal of the present note is to derive Sobolev-type regularity results for solutions to (1)–(2). Under suitable assumptions on c, h, g, the presence of the singular/degenerate homogenous weight, involving only the (N +1)-th variable t, makes the solutions to have derivates with respect to the first N variables x1, x2, . . . , xN belonging to a weighted H -space (with the same weight t1−2s); concerning the regularity of the derivative with respect to t, we obtain instead that the weighted derivative t1−2s ∂U ∂t belongs to a H-space with the dual weight t2s−1, confirming what has already been observed in [19, Lemma 7.1] for even solutions of the reflected problem corresponding to (1)–(2) with h ≡ g ≡ 0.