Paolo Caldiroli

EXISTENCE OF MINIMAL H-BUBBLES

Given a function H ∈ C1 (ℝ3) asymptotic to a constant at infinity, we investigate the existence of H-bubbles, i.e., nontrivial, conformal surfaces parametrized by the sphere, with mean curvature H. Under some global hypotheses we prove the existence of H-bubbles with minimal energy.

Singular elliptic problems with critical growth

ABSTRACT We consider Dirichlet problems of the form in Ω, u = 0 on ∂Ω, where Ω is an arbitrary domain in , with N ≥ 3, α ∈ ε(0,2), and p = 2(N−α)/(N−2) is the corresponding critical exponent. A lack of compactness may occur when or Ω is unbounded, because of concentration phenomena at the origin or vanishing, due to dilation invariance. We study the existence of positive solutions with respect to the geometry of the domain Ω.

Multiple homoclinic solutions for a class of autonomous singular systems in R2

Abstract We look for homoclinic solutions for a class of second order autonomous Hamiltonian systems in R 2 with a potential V having a strict global maximum at the origin and a finite set S ⊂ R 2 of singularities, namely V ( x ) → −∞ as dist( x , S ) → 0. We prove that if V satisfies a suitable geometrical property then for any k ∈ N the system admits a homoclinic orbit turning k times around a singularity ξ ∈ S .

Genericity of the multibump dynamics for almost periodic Duffing-like systems

In this paper we consider “slowly” oscillating perturbations of almost periodic Duffing- like systems, i.e., systems of the form ¨ u = u − (a(t) + α(ωt))W0 (u), t ∈R, u ∈RN where W ∈ C2N(RN,R) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if ω > 0 is small enough then the system admits a multibump dynamics. As a corollary we get that the system ¨ u = u−a(t)W0 (u), t ∈R, u ∈RN admits multibump solutions whenever a ∈ A, where A is an open dense subset of {a ∈ C(R,R) | a is almost periodic and a(t) > 0, ∀ t ∈R}.

Existence of H-bubbles in a perturbative setting

Given a C 1 function H:R 3 ! R, we look for H-bubbles, i.e, surfaces in R 3 parametrized by the sphere S 2 with mean curvature H at every regular point. Here we study the case H(u) = H0(u) + H 1(u) where H0 is some “good” curvature (for which there exist H0-bubbles with minimal energy, uniformly bounded in L 1 ), is the smallness parameter, and H1 is any C 1 function.

Existence and multiplicity of homoclinic orbits for potentials on unbounded domains

We study the system in R N , where V is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimising argument, we can prove the existence of a homoclinic orbit when the component Ω of {x ∈ R N : V(x) containing 0 is an arbitrary open set; in the case Ω unbounded we allow V(x) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in Ω implies that a homoclinic solution can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that the two solutions are distinct whenever the singularity is ‘not too far’ from 0.

S 2 -type Parametric Surfaces with Prescribed Mean Curvature and Minimal Energy

Given a functionHE Cl(I3) asymptotic to a constant at infinity, we investigate the existence of nontrivial, conformal surfaces parametrized by the sphere, with mean curvatureHand minimal energy.

On the existence of homoclinic orbits for the asymptotically periodic Duffing equation

where p > 1 and a : R → R satisfies: (a1) a ∈ L∞(R), inf a > 0, (a2) a = a∞ + a0, with a∞ T -periodic and a0(t)→ 0 as t→ ±∞. Noting that 0 is a hyperbolic rest point for (1.1), we look for homoclinic orbits to 0, namely non trivial solutions to (1.1) such that u(t) → 0 and u(t) → 0 as t→ ±∞. The homoclinic problem for equation (1.1), possibly with a more general nonlinearity, as well as the analogous subcritical elliptic problem on R, has been successfully studied with variational methods by several authors, for different kinds of behaviour of the coefficient a.

A Class of Second-order Dilation Invariant Inequalities

We compute the best constants in some dilation invariant inequalities for the weighted \( L^{2}-\rm {normms \; of -\Delta u \; and \nabla u } \), with weights being powers of the distance from the origin.

Symmetry Breaking of Extremals for the Caffarelli-Kohn-Nirenberg Inequalities in a Non-Hilbertian Setting

We provide an explicit necessary condition to have that no extremal for the best constant in the Caffarelli-Kohn-Nirenberg inequality is radially symmetric.