By virtue of barrier arguments we prove C α -regularity up to the boundary for the weak solutions of a non-local, non-linear problem driven by the fractional p-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular (p < 2) of degenerate (p > 2) case.Mathematics Subject Classification (2010): 35D10, 35R11, 47G20.

Abstract. We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.2000 Mathematics Subject Classification. 35P15, 35P30, 35R11.

By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth.2010 Mathematics Subject Classification. 34K37, 34B10, 46E30.

Abstract. We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C 0 -topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H s -topology.

A recent multiplicity theorem for the critical points of a functional defined on a finite-dimensional Hilbert space, established by Ricceri, is extended. An application to Dirichlet boundary value problems for difference equations involving the discrete p-Laplacian operator is presented

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