DCDS-S
Three nonzero periodic solutions for a differential inclusion
Francesca Faraci Antonio Iannizzotto
Discrete & Continuous Dynamical Systems - S 2012, 5(4): 779-788 doi: 10.3934/dcdss.2012.5.779
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
CPAA
Existence and convexity of solutions of the fractional heat equation
Antonio Greco Antonio Iannizzotto
Communications on Pure & Applied Analysis 2017, 16(6): 2201-2226 doi: 10.3934/cpaa.2017109

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
DCDS-S
Existence and multiplicity results for resonant fractional boundary value problems
Antonio Iannizzotto Nikolaos S. Papageorgiou
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 511-532 doi: 10.3934/dcdss.2018028

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

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