Three nonzero periodic solutions for a differential inclusion
Francesca Faraci Antonio Iannizzotto
We prove the existence of three non-zero periodic solutions for an ordinary differential inclusion. Our approach is variational and based on a multiplicity theorem for the critical points of a nonsmooth functional, which extends a recent result of Ricceri.
keywords: Multiplicity Variational methods. Ordinary differential inclusions Periodic solutions
Existence and multiplicity results for resonant fractional boundary value problems
Antonio Iannizzotto Nikolaos S. Papageorgiou

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.

keywords: Fractional Laplacian eigenvalue problems Morse theory
Existence and convexity of solutions of the fractional heat equation
Antonio Greco Antonio Iannizzotto

We prove that the initial-value problem for the fractional heat equation admits an entire solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. The result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al.[1] for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.

keywords: Heat equation fractional Laplacian convexity
Ground states for scalar field equations with anisotropic nonlocal nonlinearities
Antonio Iannizzotto Kanishka Perera Marco Squassina
We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.
keywords: anisotropic nonlocal nonlinearity variable exponent loss of compactness Scalar field equation existence of ground state.

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