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.

We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.2000 Mathematics Subject Classification. 35J40; 58E05. Key words and phrases. Quasi-linear Schrödinger equations, orbital stability, orbital instability by blow-up, ground state solutions, variational methods.The third author was partially supported by the Italian PRIN Research Project 2007 Metodi variazionali e topologici nello studio di fenomeni non lineari.

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.

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Abstract. We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution blows up are constructed.

Abstract. We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.

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