Search results for "multiplicity"

Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

1984

Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manif…

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

2017

Abstract In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction.We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions.These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Multiple solutions of second order Hamiltonian systems

2017

Author(s): Bonanno, G; Livrea, R; Schechter, M | Abstract: The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains

1995

Solutions and positive solutions for superlinear Robin problems

2019

We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.

A multiplicity theorem for parametric superlinear (p,q)-equations

2020

We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

Centrality and pseudorapidity dependence of the charged-particle multiplicity density in Xe–Xe collisions at sNN=5.44TeV

2019

In this Letter, the ALICE Collaboration presents the first measurements of the charged-particle multiplicity density, dNch/dη, and total charged-particle multiplicity, Nchtot, in Xe–Xe collisions at a centre-of-mass energy per nucleon–nucleon pair of sNN=5.44TeV. The measurements are performed as a function of collision centrality over a wide pseudorapidity range of −3.5<η<5. The values of dNch/dη at mid-rapidity and Nchtot for central collisions, normalised to the number of nucleons participating in the collision (Npart) as a function of sNN follow the trends established in previous heavy-ion measurements. The same quantities are also found to increase as a function of Npart, and up …

The ridge in proton-proton collisions at the LHC

2010

We show that the key features of the CMS result on the ridge correlation seen for high multiplicity events in sqrt(s)=7TeV proton-proton collisions at the LHC can be understood in the Color Glass Condensate framework of high energy QCD. The same formalism underlies the explanation of the ridge events seen in A+A collisions at RHIC, albeit it is likely that flow effects may enhance the magnitude of the signal in the latter.

Charged particle multiplicity in e^{+}e_{-}$ → q[L:q] events at 161 and 172 GeV and from the decay of the W boson

1998

The data collected by DELPHI in 1996 have been used to measure the average charged particle multiplicities and dispersions in $q\bar{q}$ events at centre-of-mass energies of $\sqrt{s}=161$~GeV and $\sqrt{s}=172$~GeV, and the average charge multiplicity in WW events at $\sqrt{s}=172$~GeV. The multiplicities in $q\bar{q}$ events are consistent with the evolution predicted by QCD. The dispersions in the multiplicity distributions are consistent with Koba-Nielsen-Olesen (KNO) scaling. The average multiplicity of charged particles in hadronic W decays has been measured for the first time; its value, $19.23 \pm 0.74 (stat+syst)$, is consistent with that expected for an $e^+e^-$ interaction at a c…

Studies of QCD at $e^{+}e^{-}$ centre-of-mass energies between 91 and 209 GeV

2004

The hadronic final states observed with the ALEPH detector at LEP in e(+)e(-) annihilation are analysed using 730 pb(-1) of data collected between 91 and 209 GeV in the framework of QCD. In particular event-shape variables and inclusive charged particle spectra are measured. The energy evolution of quantities derived from these measurements is compared to analytic QCD predictions. The mean charged particle multiplicity, the charged particle momentum spectrum and its peak position are compared to predictions of the modified-leading-logarithmic approximation. The strong coupling constant alpha(s) is determined from a fit of the QCD prediction to distributions of six event-shape variables at e…