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Gabriele Mancini

Publications and source records attributed to Gabriele Mancini.

At least 19 recordsLinked to original sources
arXiv2026

A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case

We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis--Merle. The approach we follow is based on a blow-up analysis combined with Liouville--type theorems and integral estimates. Our results extend the scalar theory of uniform a priori bounds to the Hamiltonian case, and solve an open problem in [de Figueiredo D.G., do \'O J.M., Ruf B., Adv. Nonlinear Stud. 6 (2006), no. 2]. We believe that this approach is new in this setting. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of Fixed Point Index theory.

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arXiv2025

Existence of positive solutions for a class of almost critical problems on an annulus

In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case.

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arXiv2025

A note on the first Steklov eigenvalue on planar domains

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive eigenvalue on certain symmetric domains and show that this eigenvalue is simple for all ellipses. These results complement two statements contained in a work by Kuttler and Sigillito (Proc. Amer. Math. Soc. 20, 1969).

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arXiv2024

On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality

In this paper we consider the $N$-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$, $N \geq 2$. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N \geq 2$, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant proved by Carleson and Chang for balls in $\mathbb{R}^N$.

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arXiv2024

New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D

The present survey is devoted to results on Trudinger-Moser inequalities in two dimension. We give a brief overview of the history of these celebrated inequalities and, starting from the geometric problem that motivated Moser's original work, we discuss the connection between Onofri's inequality for the unit sphere and sharp inequalities on Euclidean domains. Finally, we present recent results and new insights into nonlocal interaction energy functionals in two dimension, involving logarithmic kernels.

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arXiv2024

Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.

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arXiv2020

Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

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arXiv2019

Bubbling nodal solutions for a large perturbation of the Moser-Trudinger equation on planar domains

In this work we study the existence of nodal solutions for the problem $$ -Δu = λu e^{u^2+|u|^p} \text{ in }Ω, \; u = 0 \text{ on }\partial Ω, $$ where $Ω\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $Ω$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $Ω$ is an arbitrary domain and $λ$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser-Trudinger critical equation on a non-symmetric domain.

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arXiv2018

Local and nonlocal singular Liouville equations in Euclidean spaces

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

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arXiv2018

Glueing a peak to a non-zero limiting profile for a critical Moser-Trudinger equation

Druet [6] proved that if $(f_γ)_γ$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_γ)_γ$ solves $$ \begin{cases} &Δu =f_γ(x,u)\,,~~ u>0\text{ in }Ω\,,\\ &u =0\text{ on }\partialΩ\,, \end{cases} $$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_γ$ converges to $4πN$ plus the Dirichlet energy of $u_\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities, the loss of compactness (i.e. $N>0$) implies that $u_\infty\equiv 0$. In contrast, we prove here that there exist sequences $(f_γ)_γ$ of Moser-Trudinger type nonlinearities which admit a noncompact sequence of solutions $(u_γ)_γ$ having a nontrivial weak limit.

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arXiv2017

Improved Adams-type inequalities and their extremals in dimension 2m

In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(Ω)$, where $Ω$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams' inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

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arXiv2017

Non-existence of extremals for the Adimurthi-Druet inequality

The Adimurthi-Druet [1] inequality is an improvement of the standard Moser-Trudinger inequality by adding a $L^2$-type perturbation, quantified by $α\in [0,λ\_1)$, where $λ\_1$ is the first Dirichlet eigenvalue of $Δ$ on a smooth bounded domain. It is known [3,9,13,18] that this inequality admits extremal functions, when the perturbation parameter $α$ is small. By contrast, we prove here that the Adimurthi-Druet inequality does not admit any extremal, when the perturbation parameter $α$ approaches $λ\_1$. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler-Lagrange equation, which take into account the fact that the problem becomes singular as $α\to λ\_1$.

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arXiv2017

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega, \end{cases} \end{equation} where $h$ is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when $\Omega$ is a ball or, provided an energy control on solutions is prescribed, when $\Omega$ is a smooth bounded domain. The analogue problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.

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arXiv2017

The Moser-Trudinger inequality and its extremals on a disk via energy estimates

We study the Dirichlet energy of non-negative radially symmetric critical points $u_μ$ of the Moser-Trudinger inequality on the unit disc in $\mathbb{R}^2$, and prove that it expands as $$4π+\frac{4π}{μ^{4}}+o(μ^{-4})\le \int_{B_1}|\nabla u_μ|^2dx\le 4π+\frac{6π}{μ^{4}}+o(μ^{-4}),\quad \text{as }μ\to\infty,$$ where $μ=u_μ(0)$ is the maximum of $u_μ$. As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of the Moser-Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser-Trudinger inequality still holds, the energy of its critical points converges to $4π$ from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime.

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arXiv2016

Extremal Functions for Singular Moser-Trudinger Embeddings

We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri's inequality for the unit disk in $\mathbb{R}^2$. Moreover we consider Adimurthi-Druet type functionals on compact surfaces with conical singularities and discuss the existence of extremals for such functionals extending previous results by Castò and Roy.

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arXiv2015

Singular Liouville Equations on $S^2$: Sharp Inequalities and Existence Results

We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.

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arXiv2015

A note on compactness properties of the singular Toda system

In this note, we consider blow-up for solutions of the SU(3) Toda system on a compact surface Σ. In particular, we give a complete proof of the compactness result stated by Jost, Lin and Wang and we extend it to the case of singularities. This is a necessary tool to find solutions through variational methods.

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