arXiv subjects

Salvador López-Martínez

Publications and source records attributed to Salvador López-Martínez.

8 recordsLinked to original sources

arXiv2026

A Local Linking Theorem for Relativistic Action Functionals

We establish an analogue of the Brezis-Nirenberg local linking theorem for a class of Szulkin-type functionals arising from relativistic action principles. In this framework, compactness of Palais-Smale sequences is formulated with respect to a topology induced by the effective domain of the functional, replacing the classical strong Palais-Smale condition. The proof combines the original construction of the min-max geometry, based on a negative gradient flow, with the Ekeland-Lasry regularization. The main difficulty is that the regularized functional is naturally associated with the strong topology of the underlying functional space, whereas compactness for the original functional is formulated in the topology induced by the effective domain. We overcome this obstacle through a new perturbative construction that recovers the required min-max structure. We apply our abstract multiplicity result to two representative relativistic models: the Lorentz force equation, describing the dynamics of a charged particle in an electromagnetic field, and the Dirichlet problem for the prescribed mean curvature operator in Minkowski space. As a consequence, under natural assumptions, each problem admits at least two non-constant solutions.

math.AP↗

arXiv2026

Momentum-mass normalized dark-bright solitons to one dimensional Gross-Pitaevskii systems

We rigorously establish the existence of dark-bright solitons as traveling wave solutions to a one dimensional defocusing Gross-Pitaevskii system, a widely used model for describing mixtures of Bose-Einstein condensates and nonlinear optical systems. These solitons are shown to exhibit symmetry and radial monotonicity in modulus, and to propagate at subsonic speed. Our method relies on minimizing an energy functional subject to two constraints: the mass of the bright component and a modified momentum of the dark component. The compactness of minimizing sequences is obtained via a concentration-compactness argument, which requires some novel estimates based on symmetric decreasing rearrangements.

math.AP↗

arXiv2025

Action-minimizing periodic orbits of the Lorentz force equation with dominant vector potential

We establish the existence of non-constant periodic solutions to the Lorentz force equation, where no scalar potential is needed to induce the electromagnetic field. Our results extend to cases where a possibly singular scalar potential is present, although the vector potential assumes a leading role. The approach is based on minimizing the action functional associated with the relativistic Lagrangian. The compactness of the minimizing sequences requires the existence of negative values for the functional, which is proven using novel ideas that exploit the sign-indefinite nature of the term involving the vector potential.

math.DS↗

arXiv2025

Gray and black solitons of nonlocal Gross-Pitaevskii equations: existence, monotonicity and nonlocal-to-local limit

This article investigates the qualitative aspects of dark solitons of one-dimensional Gross-Pitaevskii equations with general nonlocal interactions, which correspond to traveling waves with subsonic speeds. Under general conditions on the potential interaction term, we provide uniform bounds, demonstrate the existence of symmetric solitons, and identify conditions under which monotonicity is lost. Additionally, we present new properties of black solitons. Moreover, we establish the nonlocal-to-local convergence, i.e. the convergence of the soliton of the nonlocal model toward the explicit dark solitons of the local Gross-Pitaevskii equation.

math.AP↗

arXiv2023

Numerical computation of dark solitons of a nonlocal nonlinear Schrödinger equation

The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross-Pitaevskii equations with nonzero boundary conditions in dimension one has been established recently in [de Laire and S. López-Martínez, Comm. Partial Differential Equations, 2022]. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. This paper provides a numerical method to compute approximations of such solitons for these types of equations, and provides actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow us to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow us to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.

math.NA↗

arXiv2022

Existence and decay of traveling waves for the nonlocal Gross-Pitaevskii equation

We consider the nonlocal Gross-Pitaevskii equation that models a Bose gas with general nonlocal interactions between particles in one spatial dimension, with constant density far away. We address the problem of the existence of traveling waves with nonvanishing conditions at infinity, i.e. dark solitons. Under general conditions on the interactions, we prove existence of dark solitons for almost every subsonic speed. Moreover, we show existence in the whole subsonic regime for a family of potentials. The proofs rely on a Mountain Pass argument combined with the so-called "monotonicity trick", as well as on a priori estimates for the Palais-Smale sequences. Finally, we establish properties of the solitons such us exponential decay at infinity and analyticity.

math.AP↗

arXiv2021

A blow-up approach for singular elliptic problems with natural growth in the gradient

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-Δu+μ(x)\frac{|\nabla u|^2}{(u+δ)^γ}= λu^p, &x\in Ω, \\ u> 0, &x\in Ω, \\ u=0, &x\in\partialΩ, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N (N\geq 3)$ is a bounded smooth domain, $λ>0$, $p>1$, $δ\geq 0$, $γ>0$ and $μ\in L^\infty(Ω)$. The main achievement resides in handling a possibly singular ($δ=0$) first order term having a nonconstant coefficient $μ$ in the presence of a superlinear zero order term. Our approach for the existence results is based on fixed point theory. With the aim of applying it, a previous analysis on a related non-homogeneous problem is carried out. The required a priori estimates are proven via a blow-up method.

math.AP↗

arXiv2019

Nonexistence result for a semilinear elliptic problem

In this paper we prove the nonexistence of nontrivial solution to \begin{equation*} \begin{cases} -Δu =f(u) &\text{in }Ω, \\ u=0 &\text{on } \partial Ω, \end{cases} \end{equation*} being $Ω\subset \mathbb{R}^N$ ($N\in \mathbb{N}$) a bounded domain and $f$ locally Lispchitz with non-positive primitive.

math.AP↗