arXiv subjects

Vittorio Coti Zelati

Publications and source records attributed to Vittorio Coti Zelati.

8 recordsLinked to original sources

arXiv2026

Asymptotic properties of non-relativistic limit for pseudo-relativistic Hartree equations

In this paper, we study the asymptotic behavior of energy and action ground states to the following pseudo-relativistic Hartree equation \[ \left(\sqrt{-c^2Δ+m^2c^4}-mc^2\right)u + λu = \left(|x|^{-1}*|u|^2\right)u \] as the speed of light $c\to\infty$. We obtain an asymptotic expansion of the ground state as $c \to \infty,$ which is new in the case of the energy ground state and generalizes the results of Choi, Hong, and Seok (2018) for the action ground state.

math.AP↗

arXiv2025

Normalized solutions for a nonlinear Dirac equation

We prove the existence of a normalized, stationary solution $Ψ\colon \mathbb{R}^{3} \to \mathbb{C}^{4}$ with frequency $w > 0$ of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form \begin{equation*} F(Ψ) = a|(Ψ, γ^{0}Ψ)|^{\fracα{2}} + b|(Ψ, γ^{1}γ^{2} γ^{3} Ψ)|^{\fracα{2}} \end{equation*} with $α\in (2,\frac{8}{3}]$, $b \geq 0$ and $a > 0$ sufficiently small. Here $γ^{i}$, $i = 0,\ldots, 3$ are the $4 \times 4$ Dirac's matrices. We find the solution as a critical point of a suitable functional restricted to the unit sphere in $L^{2}$, and $w$ turns out to be the corresponding Lagrange multiplier.

math.AP↗

arXiv2024

Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in Lorentz-Minkowski space

We prove existence of multiple radial solutions to the Dirichlet problem for nonlinear equations involving the mean curvature operator in Lorentz-Minkowski space and a nonlinear term of concave-convex type. Solutions are found using Szulkin's critical point theory for non-smooth functional. Multiplicity results are also given for some cases in which the nonlinearity depends also on the gradient of the solution.

math.AP↗

arXiv2022

Normalized solutions for the Klein Gordon-Dirac system

We prove the existence of a stationary solution for the system describing the interaction between an electron coupled with a massless scalar field (a photon). We find a solution, with fixed $L^{2}$-norm, by variational methods, as a critical point of an energy functional.

math.AP↗

arXiv2020

Traveling wave solutions to the Allen-Cahn equation

For the Allen-Cahn equation, it is well known that there is a monotone standing wave joining with the balanced wells of the potential. In this paper we study the existence of traveling wave solutions for the Allen-Cahn equation on an infinite channel. Such traveling wave solutions possess a large number of oscillation and they are obtained with the aid of variational arguments.

math.AP↗

arXiv2018

Ground state for the relativistic one electron atom

We study the Dirac-Maxwell system coupled with an external potential of Coulomb type. We use the Foldy--Wouthuysen (unitary) transformation of the Dirac operator and its realization as an elliptic problem in the 4-dim half space $\mathbb{R}^4_{+}$ with Neumann boundary condition. Using this approach we study the existence of a "ground state" solution.

math.AP↗

arXiv2014

A variational approach to the Brown-Ravenhall operator for the relativistic one-electron atoms

We use the Foldy--Wouthuysen (unitary) transformation to give an alternative characterization of the eigenvalues and eigenfunctions for the Brown-Ravenhall operator (the projected Dirac operator) in the case of a one-electron atom. In particular we transform the eigenvalues problem into an elliptic problem in the 4-dim half space $\mathbb{R}^4_{+}$ with Neumann boundary condition.

math.AP↗

arXiv2012

Ground states for pseudo-relativistic Hartree equations of critical type

We study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square-root of the Laplacian. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.

math.AP↗