Antonio Iannizzotto

Existence results for fractional p-Laplacian problems via Morse theory

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.

Weyl-type laws for fractional p-eigenvalue problems

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.

A MULTIPLICITY THEOREM FOR A PERTURBED SECOND-ORDER NON-AUTONOMOUS SYSTEM

Abstract In this paper we establish a multiplicity result for a second-order non-autonomous system.Using a variational principle of Ricceri we prove that if the set of global minima of a certain functionhas at least k connected components, then our problem has at least k periodic solutions. Moreover, theexistence of one more solution is investigated through a mountain-pass-like argument. Keywords: multiple periodic solutions; second-order non-autonomous system; critical point theory2000 Mathematics subject classification: Primary 34C25; 35A15 1. IntroductionIn this paper we consider the second-order non-autonomous system u ¨ = α ( t )( Au−∇F ( u ))+ λ∇ x G ( t,u ) , a.e. in [0 ,T ] ,u (0) −u ( T )=˙ u (0) −u ˙( T )=0 , ( P λ )where A is an N ×N symmetric matrix satisfying Ax·x c|x| 2 for all x ∈ R N , (1.1)where c is some positive constant. Assume that λ> 0, α ∈ L ∞ ([0 ,T ]), a = ess inf [0 ,T ] α> 0, F : R N → R is continuously differentiable, and that G :[0 ,T ] × R N → R is measurablein

Low-dimensional compact embeddings of symmetric Sobolev spaces with applications

If Ω is an unbounded domain in RN and p > N , the Sobolev space W 1,p(Ω) is not compactly embedded into L∞(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W 1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∞(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

Well Posed Optimization Problems and Nonconvex Chebyshev Sets in Hilbert Spaces

A result on the existence and uniqueness of metric projection for certain sets is proved, by means of a saddle point theorem. A conjecture, based on such a result and aiming for the construction of a nonconvex Chebyshev set in a Hilbert space, is presented.

A note on a question of Ricceri

Abstract Recently, Ricceri posed a question about the uniqueness of a parameter λ ∗ such that the functional x ↦ 1 2 ‖ x ‖ 2 + λ ∗ J ( x ) admits at least three critical points in a finite-dimensional Hilbert space X (under certain assumptions on the functional J ∈ C 1 ( X ) ). We give the answer, which is negative, even in the case of infinite-dimensional spaces. Moreover, we discuss the structure of the set of λ s such that the above situation occurs.

Existence of homoclinic constant sign solutions for a difference equation on the integers

We consider a difference equation involving the discrete p-Laplacian operator, depending on a positive real parameter @l. We prove, under convenient assumptions, that for @l big enough the equations admit at least one homoclinic constant sign solution in Z. Our method consists in two parts: first, we prove the existence of two Dirichlet-type solutions for the equation in the discrete interval [-n,n], for all n@?N big enough; then, we show that such solutions converge to a homoclinic solution in Z, as n->~.

Bifurcation for second order Hamiltonian systems with periodic boundary conditions

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function , and prove that the set of bifurcation points for the solutions of the system is not -compact. Then, we deal with a linear system depending on a real parameter and on a function , and prove that there exists such that the set of the functions , such that the system admits nontrivial solutions, contains an accumulation point.

Three nontrivial solutions for nonlinear fractional Laplacian equations

Abstract We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations

We consider a parametric nonlinear elliptic equation driven by the Dirichlet p-Laplacian. We study the existence, nonexistence and multiplicity of positive solutions as the parameter varies in RC 0 and the potential exhibits a p-superlinear growth, without satisfying the usual in such cases Ambrosetti–Rabinowitz condition. We prove a bifurcation-type result when the reaction has (p 1)-sublinear terms near zero (problem with concave and convex nonlinearities). We show that a similar bifurcation-type result is also true, if near zero the right hand side is (p 1)-linear.