arXiv subjects

Pietro d'Avenia

Publications and source records attributed to Pietro d'Avenia.

At least 19 recordsLinked to original sources

arXiv2026

Multi-peak solutions for a magnetic nonlinear Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$

In this paper, we study the following magnetic nonlinear Schrödinger equation \[ \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(|u|^{2})u\,\ \text{in}\,\ \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $A\in C(\mathbb{R}^{2}, \mathbb{R}^{2})$ is a magnetic potential, $V\in C(\mathbb{R}^{2}, \mathbb{R})$ is a multi-well potential, and $f\in C^1(\mathbb{R}, \mathbb{R})$ exhibits exponential critical growth. By variational methods and a penalization technique, we prove the existence and multiplicity of nontrivial solutions for sufficiently small $\varepsilon>0$. If, in addition, $A\in C^1(\mathbb{R}^2,\mathbb{R}^2)$, we show that these solutions have multiple peaks and we establish their concentration behavior and exponential decay.

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arXiv2026

Existence and Limiting Profiles of Normalized Travelling Wave Solutions for the Pseudo-Relativistic Schrödinger Equation with Logarithmic Nonlinearity

We study the existence and asymptotic behaviour of normalized solutions to the following pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \[ (\sqrt{-Δ+m^2 }-m )u+i(v\cdot \nabla )u+λu = u\log|u|^2+|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N, \] under the mass constraint \[ \|u\|_2^2=a, \] where $m,a>0$, $2<p\le\frac{2N}{N-1}$ with $N\ge 2$, $v\in \mathbb{R}^N$ is the travelling velocity with $|v|<1$, and $λ\in\mathbb{R}$ appears as Lagrange multiplier, as minima of the corresponding energy on the constraint. By applying variational method, we first provide a complete classification of the existence and nonexistence of such minima. In particular, for the mass-critical case $p=2+\frac{2}{N}$, we show that there exists a constant $a^\ast_v$ which is a threshold for the existence. Based on this, we analyse the blow-up behaviour of such minimizers as $a$ approaches $a^\ast_v$ from below. Finally, we investigate the limiting profiles of minimizers to problem when $\lim\limits_{n\to\infty}a_n=a_0\in(0,+\infty)$ with $\{a_n\}\subset(0,+\infty)$ in the mass-subcritical case $2<p<2+\frac{2}{N}$ and $\lim\limits_{n\to\infty}a_n= a_0\in(0,a^\ast_v)$ with $\{a_n\}\subset(0,a^\ast_v)$ in the mass-critical case $p=2+\frac{2}{N}$, respectively.

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arXiv2026

On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u\\ -Δϕ+a^2Δ^2ϕ=4πu^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.

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arXiv2026

Boosted Ground States for a Pseudo-Relativistic Schrödinger Equation with a double power nonlinearity

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $ψ(t,x)=e^{iλt}φ\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t ψ=(\sqrt{-Δ+m^2})ψ- |ψ|^{\frac{2}{N}}ψ-μ|ψ|^{q}ψ~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $λ$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

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arXiv2026

Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds

We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $ψ\in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.

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arXiv2026

Between Maxwell and Born-Infeld: the presence of the magnetic field

Our motivation is to consider an electromagnetic Lagrangian density $\mathcal{L}_q$, depending on a parameter such that, for $q=1$ it corresponds to the Born-Infeld Lagrangian density and for $q=2$ it restores the Maxwell one. The model in the presence of given charge and current densities is investigated. We solve the pure magnetostatic problem for $q\in(6/5,2)$. We also study the electrostatic problem in the presence of an assigned magnetic field for $q\in[1,2)$.

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arXiv2025

Normalized solutions of nonlinear magnetic Schrödinger equations on metric graphs

In this paper we first establish the theory of a magnetic Sobolev space $H^1_A(\mathcal{G},\mathbb{C})$ on metric graphs $\mathcal{G}$ and we prove the self-adjointness of its corresponding magnetic Schrödinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schrödinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.

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arXiv2025

A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.

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arXiv2025

Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities

We study the existence of solutions $(\underline u,λ_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $ψ_0$ close to $\underline u$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution.

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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.

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arXiv2024

On a fractional magnetic pseudorelativistic operator: properties and applications

We introduce a fractional magnetic pseudorelativistic operator for a general fractional order $s\in(0,1)$. First we define a suitable functional setting and we prove some fundamental properties. Then we show the behavior of the operator as $s \nearrow 1$ obtaining some results à la Bourgain-Brezis-Mironescu and removing the singularity from the integral definition. Finally we get existence of weak solutions for some semilinear equations involving a power type nonlinearity or a nonlocal (Choquard type) term.

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arXiv2023

Asymptotically linear magnetic fractional problems

The aim of this paper is investigating the existence and multiplicity of weak solutions to non--local equations involving the {\em magnetic fractional Laplacian}, when the nonlinearity is subcritical and asymptotically linear at infinity. We prove existence and multiplicity results by using variational tools, extending to the magnetic local and non--local setting some known results for the classical and the fractional Laplace operators.

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arXiv2023

Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-βΔu=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $β> -2\sqrt{m}$, where $m>0$ depends on $g$; in the latter, $N\geq 3$ and $β>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about $g$; if $N=2$ in the positive mass case, or $N=4$ in the zero mass case, we need to strengthen such assumptions. Our approach is variational.

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

Nonlinear scalar field equation with competing nonlocal terms

We find radial and nonradial solutions to the following nonlocal problem $$-Δu +ωu= \big(I_α\ast F(u)\big)f(u)-\big(I_β\ast G(u)\big)g(u) \text{ in } \mathbb{R}^N$$ under general assumptions, in the spirit of Berestycki and Lions, imposed on $f$ and $g$, where $N\geq 3$, $0\leq β\leq α 0$, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.

math.AP↗

arXiv2021

Semiclassical states for a magnetic nonlinear Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$

This paper is devoted to the magnetic nonlinear Schrödinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous functions and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ function having exponential critical growth. Under a global assumption on the potential $V$, we use variational methods and Ljusternick-Schnirelmann theory to prove existence, multiplicity, concentration, and decay of nontrivial solutions for $\varepsilon>0$ small.

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arXiv2020

Equilibrium measures and equilibrium potentials in the Born-Infeld model

In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll} -\operatorname{div}\left(\displaystyle\frac{\nabla ϕ}{\sqrt{1-|\nabla ϕ|^2}}\right)&=& ρ& \hbox{in }\mathbb{R}^N, \\[6mm] \displaystyle\lim_{|x|\to \infty}ϕ(x)&=& 0 \end{array} \right. \end{equation*} where $ρ$ is a charge distribution on the boundary of a bounded domain $Ω\subset \mathbb{R}^N$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\overline Ω$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves ($\mathcal{BI}$). Finally we characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded $C^{2,α}$-domains $Ω$ for which the equilibrium distribution is a constant multiple of the surface measure on $\partialΩ$. The same results are obtained also for Taylor approximations of the electrostatic energy.

math.AP↗

arXiv2020

Multiplicity and concentration results for a magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$

In this paper we study the following nonlinear Schrödinger equation with magnetic field \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon$ small.

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