A note on global regularity for the weak solutions of fractional p-Laplacian equations

We consider a boundary value problem driven by the fractional p-Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove H\"older regularity up to the boundary for the weak solutions, both in the singular (12) case.Comment: 7 pages, Conferenza tenuta al XXV Convegno Nazionale di Calcolo delle Variazioni, Levico 2--6 febbraio 201

Nonlocal problems at critical growth in contractible domains

We prove the existence of a positive solution for nonlocal problems involving the fractional Laplacian and a critical growth power nonlinearity when the equation is set in a suitable contractible domain.Comment: 17 page

Optimal decay of extremals for the fractional Sobolev inequality

We obtain the sharp asymptotic behavior at infinity of extremal functions for the fractional critical Sobolev embedding.Comment: 31 pages, typos correcte

Nonlocal problems with critical Hardy nonlinearity

By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.Comment: 36 pages, revised versio

Uniqueness and minimum theorems for a multifield model of brittle solids

AbstractSome minimum theorems potentially useful to construct numerical schemes related to quasi-static evolution of damage in brittle elastic solids are proposed. The approach is that of multifield theories, with a second-order damage tensor describing the microcrack density. The use of damage entropy flux and damage pseudo-potential are both investigated

The Brezis-Nirenberg problem for the fractional pp-Laplacian

We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional $p$-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when $p \ne 2$. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained by Brasco, Mosconi and Squassina. The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking theorem. We use an abstract linking theorem based on the cohomological index proved by Perera and Yang to overcome this difficulty.Comment: 24 page

Recent progresses in the theory of nonlinear nonlocal problems

We overview some recent existence and regularity results in the theory of nonlocal nonlinear problems driven by the fractional p-Laplacian.We overview some recent existence and regularity results in the theory of nonlocal nonlinear problems driven by the fractional p-Laplacian