Francesca Faraci

Infinitely many periodic solutions for a second-order nonautonomous system

In this paper we are interested in multiplicity results for a second-order nonautonomous system. Infinitely many solutions follow from a recent variational principle by B. Ricceri.

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.

On Stochastic Variational Inequalities with Mean Value Constraints

In this note, we consider a class of variational inequalities on probabilistic Lebesgue spaces, where the constraints are satisfied on average, and provide an approximation procedure for the solutions. As an application, we investigate the Nash–Cournot oligopoly problem with uncertain data and compare the solutions obtained when the constraints are satisfied on average with the ones obtained when the constraints are satisfied almost surely.

ON AN OPEN QUESTION OF RICCERI CONCERNING A NEUMANN PROBLEM

In this paper we solve partially an open problem raised by B. Ricceri ( Bull. London Math. Soc. 33 (2001), 331–340). Infinitely many solutions for a Neumann problem are obtained through a direct variational approach where the nonlinearity has an oscillatory behaviour at infinity.

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.

BOUNDED MULTIPLE SOLUTIONS FOR -LAPLACIAN PROBLEMS WITH ARBITRARY PERTURBATIONS

In the present paper we deal with the existence of multiple solutions for a quasilinear elliptic problem involving an arbitrary perturbation. Our approach, based on an abstract result of Ricceri, combines truncation arguments with Moser-type iteration technique.

On a singular semilinear elliptic problem: multiple solutions via critical point theory

We study existence and multiplicity of solutions of a semilinear elliptic problem involving a singular term. Combining various techniques from critical point theory, under different sets of assumptions, we prove the existence of

An Overview on Singular Nonlinear Elliptic Boundary Value Problems

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