The effect of the Hardy potential in some Calderón–Zygmund properties for the fractional Laplacian

Abstract

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problemsin Ω,Lipschitz boundary such that 0 ∈ Ω and N > 2s. We will mainly consider the solvability in two cases:(1) The linear problem, that is, f (x, t) = f (x), where according to the summability of the datum f and the parameter λ we give the summability of the solution u.(2) The problem with a nonlinear term f (x, t) = h(x) t σ for t > 0. In this case, existence and regularity will depend on the value of σ and on the summability of h. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.

Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

“…Abdellaoui and Medina et al in [13] gave the solvability of the problem (2) for the linear case g(x, t) = g(x) and the nonlinear case g(x, t) = h(x) t σ , respectively. For critical case, a positive solution was obtained in [14] with by the Lagrange multipliers technique.…”

Combined effects of the Hardy potential and lower order terms in fractional Laplacian equations

“…Proof The complete proof is given in [12], for the reader's convenience, we include here a sketch of the proof.…”

“…Recently a great attention has been devoted to understanding the role of the Hardy potential in the solvability of fractional elliptic problem; see for instance [9][10][11][12][13] and the references therein. In particular, Abdellaoui et al [12] obtained regularity of solutions to the following nonlocal nonlinear problem:…”

“…In particular, Abdellaoui et al [12] obtained regularity of solutions to the following nonlocal nonlinear problem:…”

“…The influence of the Hardy potential for fractional Laplacian was studied in [12], the main effect of the Hardy potential in (1.3) is that the weak solutions to problem (1.3) satisfy u(x) ≥ C|x| -γ for some constants C and γ , this fact shows that u(x) is unbounded in a neighborhood of the origin, instead of u(x) ∈ L ∞ ( ). On the other hand, it is well known that the lower order term u p produces a regularizing effect; see [14][15][16][17] and the references therein.…”