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Marco G. Ghimenti

Publications and source records attributed to Marco G. Ghimenti.

18 recordsLinked to original sources
arXiv2023

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

math.AP↗
arXiv2022

Multiple solutions and profile description for a nonlinear Schrödinger-Bopp-Podolsky-Proca system on a manifold

We prove a multiplicity result for \begin{equation*} \begin{cases} -\varepsilon^{2}Δ_g u+ωu+q^{2}ϕu=|u|^{p-2}u\\[1mm] -Δ_g ϕ+a^{2}Δ_g^{2} ϕ+ m^2 ϕ=4πu^{2} \end{cases} \text{ in }M, \end{equation*} where $(M,g)$ is a smooth and compact $3$-dimensional Riemannian manifold without boundary, $p\in(4,6)$, $a,m,q\neq 0$, $\varepsilon>0$ small enough. The proof of this result relies on Lusternik-Schnirellman category. We also provide a profile description for low energy solutions.

math.AP↗
arXiv2021

Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.

math.AP↗
arXiv2020

A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n=6,7,8, provided that the Weyl tensor is always not vanishing on the boundary.

math.DG↗
arXiv2010

Periodic orbits of a one dimensional non autonomous Hamiltonian system

In this paper we study the properties of the periodic orbits of x + V'_x(t, x) = 0 with x \in S1 and V(t, x) a T0 periodic potential. Called ρ \in (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.

math.CA↗
arXiv2010

Semiclassical limit for the nonlinear Klein Gordon equation in bounded domains

We are interested to the existence of standing waves for the nonlinear Klein Gordon equation ε^2{\box}ψ + W'(ψ) = 0 in a bounded domain D. The main result of this paper is that, under suitable growth condition on W, for ε sufficiently small, we have at least cat(D) standing wavesfor the equation (†), while cat(D) is the Ljusternik-Schnirelmann category.

math.AP↗
arXiv2010

The Nonlinear Schroedinger Equation: Existence, Stability and Dynamics of Solitons

In this paper we present some recent results concerning the ex- istence, the stability and the dynamics of solitons occurring in the nonlinear Schroedinger equation when the parameter h -> 0. We focus on the role played by the Energy and the Charge in the existence, the stability and the dynamics of solitons. Moreover, we show that, under suitable assumptions, the soliton approximately follows the dynamics of a point particle.

math.AP↗
arXiv2010

Positive solutions of singularly perturbed nonlinear elliptic problem on Riemannian manifolds with boundary

Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -ε^2Δ_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partialν=0 on \partial M where ν is an exterior normal to \partial M. The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.

math.AP↗
arXiv2010

Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for (ε, h) belonging to a residual subset of (0, ε0) \times Bρ, for (ε0, ρ) small enough.

math.AP↗
arXiv2010

Geodesics in Conical Manifolds

The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of $\R^n$ with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the calculus of variation in the large

math.AP↗
arXiv2010

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. We define these manifolds as submanifolds of $\R^n$ with a finite number of conical singularities. To formulate a good Morse theory we must use an appropriate definition of geodesic. The main theorem of this paper claims that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful.

math.AP↗