This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-\Delta)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

PB - University of Catania U1 - 7318 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Asymptotics of the s-perimeter as s →0 JF - Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790 Y1 - 2012 A1 - Serena Dipierro A1 - Alessio Figalli A1 - Giampiero Palatucci A1 - Enrico Valdinoci AB -We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

PB - American Institute of Mathematical Sciences U1 - 7317 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - On periodic elliptic equations with gradient dependence JF - Communications on Pure and Applied Analysis Y1 - 2008 A1 - Massimiliano Berti A1 - Matzeu, M A1 - Enrico Valdinoci AB - We construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope. VL - 7 N1 - cited By (since 1996)1 ER - TY - JOUR T1 - Periodic orbits close to elliptic tori and applications to the three-body problem JF - Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138 Y1 - 2004 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Enrico Valdinoci AB - We prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.) PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/2985 U1 - 1348 U2 - Mathematics U3 - Functional Analysis and Applications ER -