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Teresa D'Aprile

Publications and source records attributed to Teresa D'Aprile.

15 recordsLinked to original sources
arXiv2026

Higher-order criteria and sharpness for blow-up at quantized singularities in Liouville equations

This work is devoted to the study of blow-up phenomena for solutions of \[-Δu=λV(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $λ>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $λ\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.

math.AP↗
arXiv2026

Second order classification for singular Liouville equations with a coefficient function

In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-Δv= λV(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, and $λ>0$ is a small parameter. We find necessary and sufficient conditions on the potential $V$ for the existence of a blow-up sequence of solutions tending to infinity near the origin as $λ\to 0^+$. In particular, we obtain a second-order classification of the coefficient function $V$ for which (simple) blow-up occurs at the origin.

math.AP↗
arXiv2024

On Non-simple blowup solutions of Liouville equation

For Liouville equation with quantized singular sources, the non-simple blowup phenomenon has been a major difficulty for years. It was conjectured by the first two authors that the non-simple blowup phenomenon does not occur if the equation is defined on the unit ball with Dirichlet boundary condition. In this article we not only completely settle this conjecture in its entirety, but also extend our result to cover any bounded domain. Since the main theorem in this article rules out the non-simple phenomenon in commonly observed applications, it may pave the way for advances in degree counting programs, uniqueness of blowup solutions and construction of solutions, etc.

math.AP↗
arXiv2024

High Order Vanishing Theorems for Nonsimple Blowup Solutions of Singular Liouville Equation

For a singular Liouville equation, it is plausible that a non-simple blowup phenomenon occurs around a quantized singular pole. The presence of complex blowup profiles of bubbling solutions presents substantial challenges in applications. In this article, we demonstrate that under natural assumptions, non-simple blowup takes place only when the derivatives of certain coefficient functions approach zero. Our main result encompasses all previous findings and determines the vanishing order for any specific quantized singular source. Our theorems can be utilized not only to eliminate multiple non-simple blowup scenarios in applications but also to investigate blowup solutions with moving poles.

math.AP↗
arXiv2023

On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-Δu= λV(x) e^u-4πN δ_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, $N$ is a positive integer ($N\geq 1$). Here $δ_0$ defines the Dirac measure with pole at $0$, and $λ>0$ is a small parameter. We assume that $N=1$ and, under some suitable assumptions on the derivatives of the potential $V$ at $0$, we find a solution which exhibits a non-simple blow-up profile as $λ\to 0^+$.

math.AP↗
arXiv2021

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-Δu= \la a(x) e^u-4πN δ_0\;\hbox{ in } Ω,\quad u=0 \;\hbox{ on }\partial Ω,$$ where $Ω$ is a smooth and bounded domain in $\R^2$ such that $0\inΩ$, $a(x)$ is a positive smooth function, $N$ is a positive integer and $\la>0$ is a small parameter. Here $δ_0$ defines the Dirac measure with pole at $0$. We find conditions on the function $a$ and on the domain $Ω$ under which there exists a solution $u_\la$ blowing up at $0$ and satisfying $\la\into a(x)e^{u_\la} \to 8π(N+1)$ as $\la\to 0^+$.

math.AP↗
arXiv2017

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $Σ$ admitting conical singularities of orders $α_i$'s at points $p_i$'s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $χ(Σ)+\sum_i α_i$ approaches a positive even integer, where $χ(Σ)$ is the Euler characteristic of the surface $Σ$.

math.AP↗
arXiv2015

Equilibria of point-vortices on closed surfaces

We discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface $Σ$. The topological properties of $Σ$ determine the occurrence of three distinct situations, corresponding to $\mathbb{S}^2$, to $\mathbb{RP}^2$ and to $Σ\not=\mathbb{S}^2,\mathbb{RP}^2$. As a by-product, we also obtain new existence results for the singular mean-field equation with exponential nonlinearity.

math.AP↗
arXiv2014

Asymmetric blow-up for the SU(3) Toda System

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

math.AP↗
arXiv2012

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem {equation*}\e^2Δv-v+f(v)=0\hbox{in}Ω,\quad v=0 \hbox{on}\partial Ω,{equation*} where $Ω$ is a smooth and bounded domain of $\R^N$, $\e$ is a small positive parameter, $f$ is a superlinear, subcritical and odd nonlinearity. In particular we prove that if $Ω$ has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that $k$ is even and sufficiently large, a $k$-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of $\partial Ω$.

math.AP↗
arXiv2012

Multiple blow-up solutions for the Liouville equation with singular data

We study the existence of solutions with multiple concentration to the following boundary value problem $$-Δu=\e^2 e^u-4π\sum_{p\in Z}α_p δ_{p}\;\hbox{in} Ω,\quad u=0 \;\hbox{on}\partial Ω,$$ where $Ω$ is a smooth and bounded domain in $\R^2$, $α_{p}$'s are positive numbers, $Z\subset Ω$ is a finite set, $δ_p$ defines the Dirac mass at $p$, and $\e>0$ is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso (\cite{delkomu}) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights $α_p$, a solution exists with a number of blow-up points $ξ_j\in Ω\setminus Z$ up to $\sum_{p\in Z}\max\{n\in\N\,|\, n<1+α_p\}$.

math.AP↗
arXiv2012

Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-Δu=|u|^{2^*-2-\e}u \hbox{in}Ω, \quad u=0 \hbox{on}\partial Ω,$$ where $Ω$ is a smooth bounded domain in $\R^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\e>0$ is a small parameter. In particular we prove that if $Ω$ is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.

math.AP↗
arXiv2012

On the profile of sign changing solutions of an almost critical problem in the ball

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\eps>0$ is a small parameter. Using a Lyapunov-Schmidt reduction we discover two new non-radial solutions having 3 bubbles with different nodal structures. An interesting feature is that the solutions are obtained as a local minimum and a local saddle point of a reduced function, hence they do not have a global min-max description.

math.AP↗
arXiv2009

Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

We study the existence of positive and sign-changing multipeak solutions for the stationary Nonlinear Schroedinger Equation. Here no symmetry on $V$ is assumed. It is known that this equation has positive multipeak solutions with all peaks approaching a local maximum of the potential. It is also proved that solutions alternating positive and negative spikes exist in the case of a minima. The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of the external potential.

math.AP↗