Soliton dynamics for a general class of Schr\"odinger equations

The soliton dynamics for a general class of nonlinear focusing Schr\"odinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.Comment: 26 page

H^s versus C^0-weighted minimizers

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.Comment: 15 page

Weak and viscosity solutions of the fractional Laplace equation

Aim of this paper is to give a regularity result for weak solutions of a fractional Laplacian equation. In order to get this result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem. As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of the fractional Laplacian is strictly positive

An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

Asymptotically linear problems driven by fractional Laplacian operators

In this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one

A critical fractional equation with concave-convex power nonlinearities

none4sìIn this work we study a fractional critical problem with concave-convex nonlinearities. Our main results show the existence and multiplicity of solutions to this problem for different values of the real parameter appearing in the equation. The dependency on this parameter changes according to whether we consider the concave power case or the convex power case. These two cases will be treated separatelyopenBarrios B; Colorado E; Servadei R; Soria FBarrios, B; Colorado, E; Servadei, Raffaella; Soria, F