Sharp lower bounds for Coulomb energy
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $\dot H^s(\R^3)$ with $\frac 12 <s<\frac{3}{2}$. In case $\frac 12 <s \leq 1$ we show that the lower bounds are sharp.
math.AP↗
arXiv subjects
Marco Ghimenti
Publications and source records attributed to Marco Ghimenti.
Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.
math.AP↗
Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary
Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.
math.AP↗
Number and profile of low energy solutions for singularly perturbed Klein Gordon Maxwell systems on a Riemannian manifold
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of the nonlinear Klein-Gordon-Maxwell system and nonlinear Schroedinger-Maxwell system with subcritical nonlinearity. We prove that the number of one peak solutions depends on the topological properties of the manifold M, by means of the Lusternik Schnirelmann category.
math-ph↗
Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold
Given a smooth Riemannian manifold (M,g)we investigate the existence of positive solutions to a singularly perturbed supercritical elliptic equation which concentrate at some submanifold of M. We obtain a posive answer for some manifolds, which include warped products.
math.AP↗
Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold
We establish the existence of semiclassical states for a nonlinear Klein-Gordon-Maxwell-Proca system in static form, with Proca mass 1 on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of 3-dimensional manifolds on which the system has semiclassical solutions for every exponent p>2. The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero
math.AP↗
Blowing-up solutions concentrating along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds
Let $(M,g)$ and $(K,κ)$ be two Riemannian manifolds of dimensions $m$ and $k ,$ respectively. Let $ω\in C^2(N),$ $ω> 0.$ The warped product $ M\times _ωK$ is the $ (m+k)$-dimensional product manifold $M\times K$ furnished with metric $ g+ω^2 κ.$ We prove that the supercritical problem $$-Δ_{g+ω^2 κ}u+h u=u^{ {m+2\over m-2} \pm\varepsilon},\ u>0,\ \hbox{in}\ (M\times _ωK,g+ω^2 κ)$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $Γ$ of $M\times _ωN$ as the real parameter $\varepsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ satisfy a suitable condition.
math.AP↗
Soliton dynamics for the generalized Choquard equation
We investigate the soliton dynamics for a class of nonlinear Schrödinger equations with a non-local nonlinear term. In particular, we consider what we call {\em generalized Choquard equation} where the nonlinear term is $(|x|^{θ-N} * |u|^p)|u|^{p-2}u$. This problem is particularly interesting because the ground state solutions are not known to be unique or non-degenerate.
math.AP↗
Multi-solitary waves for the nonlinear Klein-Gordon equation
We consider the nonlinear Klein-Gordon equation in $\R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
math.AP↗
Nondegeneracy of critical points of the mean curvature of the boundary for Riemannian manifolds
Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $\partial M$and $\partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of the mean curvature of $\partial M$ are nondegenerate.
math.AP↗
Low energy solutions for the semiclassical limit of Schroedinger Maxwell systems
We show that the number of solutions of Schroedinger Maxwell system on a smooth bounded domain in R^3 depends on the topological properties of the domain. In particular we consider the Lusternik-Schnirelmann category and the Poincaré polynomial of the domain.
math-ph↗
The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of singularly perturbed Klein-Gordon-Maxwell systems and Schroedinger-Maxwell systems on M, with a subcritical nonlinearity. We prove that when the perturbation parameter epsilon is small enough, any stable critical point x_0 of the scalar curvature of the manifold (M,g) generates a positive solution (u_eps,v_eps) to both the systems such that u_eps concentrates at xi_0 as epsilon goes to zero.
math-ph↗
Soliton dynamics of NLS with singular potentials
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrödinger equation in $\R^{N}, N\ge 3$, in presence of a singular external potential.
math.AP↗
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
math.AP↗
Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number ε and the symmetric metric g. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
math.AP↗
Some generic properties of non degeneracy for critical points of functionals and applications
We give some generic properties of non degeneracy for critical points of functionals. We apply these results, obtaining some theorems of multiplicity of solutions for the equation -ε^2Δ_g u+u=|u|p-2u in M, u in H_g^1(M) where M is a compact Riemannian manifold of dimension n and 2< p<2n/(n-2).
math.AP↗
On the Dynamics of solitons in the nonlinear Schroedinger equation
We study the behavior of the soliton solutions of the equation i((\partialψ)/(\partialt))=-(1/(2m))Δψ+(1/2)W_{ε}'(ψ)+V(x)ψ where W_{ε}' is a suitable nonlinear term which is singular for ε=0. We use the "strong" nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for ε\to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).
math-ph↗
An elementary approach to Stochastic Differential Equations using the infinitesimals
The aim of this paper is to evidence two points relative to Nonstandard analysis (NSA): 1. In most applications of NSA to analysis, only elementary facts and techniques of nonstandard calculus seems to be necessary. 2. The advantages of a theory which includes infinitesimals rely more on the possibility of making new models rather than in the proving techniques. These two points will be illustrated in the theory of Brownian motion which can be considered as a classical model to test the power of the infinitesimal approach. Starting from a naive idea of Brownian motion, we deduce the Fokker-Plank equation in a simple and rigorous way. It is possible to keep every things to a simple level since all the theory of stochastic differential equations is treated as a hyperfinite theory and it is not translated in a "standard model". The only standard object is the final one: the Fokker-Plank equation.
math-ph↗