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Antonio Azzollini

Publications and source records attributed to Antonio Azzollini.

At least 19 recordsLinked to original sources
arXiv2025

Schrödinger equation in dimension two with competing logarithmic self-interaction

In this paper we study the equation \[ -Δu +(\log |\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2, \] where $8/3 < q < 4$. By means of variational arguments, we find infinitely many radially symmetric classical solutions. The main difficulties rely on the competition between the two nonlocal terms and on the presence of logarithmic kernels, which have not a prescribed sign. In addition, in order to find finite energy solutions, a suitable functional setting analysis is required.

math.AP↗
arXiv2021

The 2-dimensional nonlinear Schrodinger-Maxwell system

In this paper we carry on the study of a system recently introduced by the first author as the planar version of the well known electrostatic Schrödinger - Maxwell equations. In the positive potential case, we exhibit situations where the existence of solutions depends on the strength of the coupling, being this one modulated by a parameter. We also present some results in the case of a sign-changing potential.

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arXiv2020

Finite energy standing waves for the Klein-Gordon-Maxwell system: the limit case

In this paper we consider the Klein-Gordon-Maxwell system in the electrostatic case, assuming the fall-off large-distance requirement on the gauge potential. We are interested in proving the existence of finite energy (and finite charge) standing waves, having the phase corresponding to the mass coefficient in the Klein-Gordon Lagrangian.

math.AP↗
arXiv2020

The planar Schrödinger-Poisson system with a positive potential

In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -Δu \pm ϕu + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline Δϕ= u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$ is assumed positive. In dimension three, the problem with the sign + (we call it $(\mathcal P_+)$) was considered and solved in \cite{M}, whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign -- (say it $(\mathcal P_-)$). We provide a general existence result for $(\mathcal P_+)$ and two examples falling in the case $(\mathcal P_-)$ for which there exists at least a nontrivial solution.

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arXiv2017

On the Schrödinger-Born-Infeld system

In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrödinger and the Born-Infeld lagrangians, this latter replacing the role that, classically, is played by the Maxwell lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.

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arXiv2016

Generalized Schrödinger-Newton system in dimension $N\ge 3$: critical case

In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the nonresonance case for higher dimensions.

math.AP↗
arXiv2015

On a prescribed mean curvature equation in Lorentz-Minkowski space

We are interested in providing new results on a prescribed mean curvature equation in Lorentz-Minkowski space set in the whole R^N, with N >2. We study both existence and multiplicity of radial ground state solutions for p>1, emphasizing the fundamental difference between the subcritical and the supercritical case. We also study speed decay at infinity of ground states, and give some decay estimates. Finally we provide a multiplicity result on the existence of sign-changing bound state solutions for any p>1.

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arXiv2014

A multiplicity result for the nonlinear Klein Gordon Maxwell equations

In this paper we provide a new technique to find solutions to the Klein-Gordon-Maxwell system. The method, based on an iterative argument, permits to improve previous results where the reduction method was used. We also show how this device permits to obtain a multiplicity result in the physically significant context known as "the positive potential case".

math.AP↗
arXiv2011

On a system involving a critically growing nonlinearity

This paper deals with the system \[\{{array}{ll} -Δu = λu + q |u|^3 u ϕ& \hbox{in} B_R, -Δϕ=q |u|^5 & \hbox{in} B_R, u=ϕ=0 & \hbox{on} \partial B_R. {array}.\] We prove existence and nonexistence results depending on the value of $λ$.

math.AP↗
arXiv2011

The elliptic Kirchhoff equation in $\R^N$ perturbed by a local nonlinearity

In this paper we present a very simple proof of the existence of at least one non trivial solution for a Kirchhoff type equation on $\RN$, for $N\ge 3$. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.

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arXiv2010

Multiple critical points for a class of nonlinear functionals

In this paper we prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrodinger-Maxwell system and to the nonlinear elliptic Kirchhoff equation assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.

math.AP↗
arXiv2009

Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system

We study the class of nonlinear Klein-Gordon-Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well known "zero mass case" for scalar field equations.

math.AP↗