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Pietro d'Avenia

Publications and source records attributed to Pietro d'Avenia.

At least 37 records · Page 2Linked to original sources
arXiv2020

Oscillating solutions for nonlinear equations involving the Pucci's extremal operators

This paper deals with the following nonlinear equations \[ \mathcal{M}_{λ,Λ}^\pm(D^2 u)+g(u)=0 \qquad \hbox{ in }\mathbb{R}^N, \] where $\mathcal{M}_{λ,Λ}^\pm$ are the Pucci's extremal operators, for $N \ge 1$ and under the assumption $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.

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arXiv2018

Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

We study the following nonlinear Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu + ωu + q^{2}ϕu = |u|^{p-2}u -Δϕ+ a^2 Δ^2 ϕ= 4πu^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,ω>0$. We prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schrödinger-Poisson system as $a\to0$.

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arXiv2017

Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^N$ are continuous potentials and $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\varepsilon$ small.

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arXiv2016

Ground states for fractional magnetic operators

We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.

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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.

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arXiv2016

Vortex ground states for Klein-Gordon-Maxwell-Proca type systems

We look for three dimensional vortex-solutions, which have finite energy and are stationary solutions, of Klein-Gordon-Maxwell-Proca type systems of equations. We prove the existence of three dimensional cylindrically symmetric vortex-solutions having a least possible energy among all symmetric solutions. Moreover we show that, if the Proca mass disappears, then the solutions tends to a solution of the Klein-Gordon-Maxwell system.

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arXiv2015

On the electrostatic Born-Infeld equation with extended charges

In this paper, we deal with the electrostatic Born-Infeld equation \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{ \begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla ϕ}{\sqrt{1-|\nabla ϕ|^2}}\right)= ρ, & \hbox{in } \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}ϕ(x)= 0, \end{array} \right. \end{equation} where $ρ$ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential $ϕ$ and finiteness of the energy of the electrostatic field $-\nabla ϕ$. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming $ρ$ is radially distributed, we recover the weak formulation of \eqref{eq:BI-abs} and the regularity of the solution of the Poisson equation (under the same smootheness assumptions). In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if $ρ$ is smooth. Then we analyze the case where the density $ρ$ is a superposition of point charges and discuss the results in [Kiessling, Comm. Math. Phys. 314 (2012), 509--523]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.

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arXiv2014

Fractional logarithmic Schrödinger equations

By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions.

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arXiv2014

On fractional Choquard equations

We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.

math.AP↗
arXiv2014

A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity

In this paper we give a multiplicity result for the following Chern-Simons-Schrödinger equation \[ -Δu+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s τu^2(τ) \ d τ$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.

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arXiv2014

Positive ground states for a system of Schrödinger equations with critically growing nonlinearities

We study the following problem \[ \begin{cases} -Δu = λu + u^{2^*-2} v & \hbox{in} Ω,\\ -Δv= μv^{2^*-1} + u^{2^*-1} & \hbox{in} Ω,\\ u> 0,v> 0 & \hbox{in} Ω,\\ u=v=0 & \hbox{on} \partial Ω, \end{cases} \] where $Ω$ is a bounded domain of $\mathbb{R}^N$, $N\geq 4$, $2^*=2N/(N-2)$, $λ\in\mathbb{R}$ and $μ\geq 0$ and we obtain existence and nonexistence results, depending on the value of the parameters $λ$ and $μ$.

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arXiv2013

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↗
arXiv2013

On the logarithmic Schrodinger equation

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

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