1,311 research outputs found
Scaling properties of functionals and existence of constrained minimizers
No full textIn this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on Hilbert spaces that are invariant under translations. Our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals. We introduce family of maps, called scaling paths, that permits to show the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional I associated to the Schrödinger-Poisson equation in R3iΨt+δΨ-(|x|-1*|Ψ|2)Ψ+|Ψ|p-2Ψ=0 when 2<3. In particular we prove that I achieves its minimum on the constraint {u∈H1(R3):||u||2=Ρ} for every sufficiently small Ρ>0. In this way we recover the case studied in Sanchez and Soler (2004) [20] for p=8/3 and we complete the case studied by the authors for 3<10/3 in Bellazzini and Siciliano (2011) [4]. © 2011 Elsevier Inc
Stable standing waves for a class of nonlinear Schrödinger-Poisson equations
No full textWe prove the existence of orbitally stable standing waves with prescribed L2-norm for the following Schrödinger-Poisson type equation in R when p = {8/3} ∪(3, 10/3).In the case 3 < p < 10/3, we prove the existence and stability only for sufficiently large L2-norm. In case p = 8/3, our approach recovers the result of Sanchez and Soler (J Stat Phys 114:179-204, 2004) for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In the final section, a further application to the Schrödinger equation involving the biharmonic operator is given. © 2010 Springer Basel AG
On the orbital stability for a class of nonautonomous NLS
No full textFollowing the original approach introduced by T. Cazenave and P.L. Lions in [4] we prove the existence and the orbital stability of standing waves for the
following class of NLS:
(0.1)
i@tu + u − V (x)u + Q(x)u|u|p−2 = 0, (t, x) 2 R × Rn, 2 < p < 2 + 4/n
and
(0.2) i@tu − Δ2u − V (x)u + Q(x)u|u|p−2 = 0, (t, x) 2 R × Rn, 2 < p < 2 + 8/n
under suitable assumptions on the potentials V (x) and Q(x).
More precisely we assume V (x),Q(x) 2 L∞(Rn) and meas{Q(x) < λ0} ε (0,∞) for a suitable λ0<0. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to (0.1) and (0.2)
Max-Min characterization of the mountain pass energy level for a class of variational problems
No full textWe prove the existence of a critical point at the mountain pass energy level for a general class of variational problems. We also provide a Max-Min characterization of the mountain pass energy level. Finally, we present some concrete applications
Nonlinear Schroedinger equations with strongly singular potentials
No full textIn this paper we look for standing waves for nonlinear
Schr\"odinger equations
with cylindrically symmetric potentials vanishing at infinity and non-increasing,
and a nonlinear term satisfying weak assumptions. In
particular we show the existence of standing waves with
non-vanishing angular momentum with prescribed norm. The
solutions are obtained via a minimization argument, and the proof
is given for an abstract functional which presents lack of
compactness. As a particular case we prove the existence of standing waves with
non-vanishing angular momentum for the nonlinear hydrogen atom equation
Periodic orbits of a one-dimensional non-autonomous Hamiltonian system
No full textIn this paper we study the properties of the periodic orbits of [...] + V 'x (t, x) = 0 with x∈S1 and V 'x (t, x) a T0 periodic potential. Called ρ∈1/T0 Q the frequency of windings of an orbit in S1 we show that exists an infinite number of periodic solutions with a given ρ. We give a lower bound on the number of periodic orbits with a given period and ρ by means of the Morse theory
A study of complexity in gamma ray burst using the diffusion entropy approach
No full textThe Diffusion Entropy algorithm is a method that allows the study of correlated non-stationary time series and allows the discrimination between signal and uncorrelated noise.
DE provides a quantitative measure of the complexity by means of a scaling index δ.
The aim of this paper is to apply this method to study and statistically characterize Gamma-Ray Burst light curves and to introduce a method to constrain and test GRB models
Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems.
No full textIn this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma
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