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Eugenio Vecchi

Publications and source records attributed to Eugenio Vecchi.

At least 19 recordsLinked to original sources
arXiv2026

On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''

We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. The main novelty is that we consider a mixed operator of the form $-Δ- γ(-Δ)^s$, namely we suppose that the fractional Laplacian has the ``wrong sign'' and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.

math.AP↗
arXiv2026

Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data

We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural hypotheses, the prototype being $f(u)=u^p$ with $p>1$. By adapting the Kaplan method to the present framework, we prove that solutions blow up in finite time whenever the initial datum is sufficiently large. In the prototype case $f(u)=u^p$, this conclusion holds for every $p>1$. As a particular case of our operator, we also include the fractional Laplacian; to the best of our knowledge, this type of result is new even in that special case.

math.AP↗
arXiv2026

Critical concave-convex problems in Carnot groups

We consider a model Dirichlet problem with concave-convex and critical nonlinearity settled in Carnot groups. Our aim is to prove the existence of two positve solutions in the spirit of a famous result by Ambrosetti, Brezis and Cerami. To this aim we use a variational Perron method combined with proper estimates of a family of functions which are minimizers of the relevant Sobolev inequality. Due to the lack of boundary regularity, we also have to be careful while proving that the first solution found is a local minimizer in the proper topology.

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arXiv2025

Global weak solutions for the inverse mean curvature flow in the Heisenberg group

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\varepsilon=0$. For $Ω\subseteq \He^n$ an open set with smooth boundary $Σ_0=\partial Ω$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{Σ^\varepsilon_s\}_{s \geq 0} \subseteq \mathbb{H}^n$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to Moser and based on the the link between IMCF and $p$-harmonic functions.

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arXiv2025

Critical singular problems in Carnot groups

We consider a power-type mild singular perturbation of a Dirichlet semilinear critical problem settled in an open and bounded set in a Carnot group. Here, the term critical has to be understood in the sense of the Sobolev embedding. We aim to prove the existence of two positive weak solutions: the first one is obtained by means of the variational Perron's method, while for the second one we adapt a classical argument relying on proper estimates of a family of functions which mimic the role of the classical Aubin-Talenti functions in the Euclidean setting. Our results fall in the framework of semilinear PDEs in Carnot group but, as far as we know, are the first ones dealing with singular perturbations of power-type.

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arXiv2025

A Brezis-Nirenberg type result for mixed local and nonlocal operators

We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. In particular, we investigate the corresponding Sobolev inequality, detecting the optimal constant, which we show that is never achieved. Moreover, we present an existence (and nonexistence) theory for the corresponding subcritical perturbation problem.

math.AP↗
arXiv2024

Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian

In this paper we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic $p$-Laplacian. The critical exponent is the usual $p^{\star}$ such that the embedding $W^{1,p}_{0}(Ω) \subset L^{p^{\star}}(Ω)$ is not compact. We prove the existence of a weak positive solution in presence of both a $p$-linear and a $p$-superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin-Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis-Nirenberg \cite{BN}.

math.AP↗
arXiv2024

On mixed local-nonlocal problems with Hardy potential

In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-Δu + (-Δ)^s u - γ\frac{u}{|x|^2}=f \text{ in } Ω, \ u=0 \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $γ> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $γ$.

math.AP↗
arXiv2024

Multiplicity of positive solutions for mixed local-nonlocal singular critical problems

We prove the existence of at least two positive weak solutions for mixed local-nonlocal singular and critical semilinear elliptic problems in the spirit of [Haitao, 2003], extending the recent results in [Garain, 2023] concerning singular problems and, at the same time, the results in [Biagi, Dipierro, Valdinoci, Vecchi, 2022] regarding critical problems.

math.AP↗
arXiv2023

An Ahmad-Lazer-Paul-type result for indefinite mixed local-nonlocal problems

We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eigenvalues.

math.AP↗
arXiv2022

A Faber-Krahn inequality for mixed local and nonlocal operators

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.

math.AP↗
arXiv2022

A Brezis-Oswald approach for mixed local and nonlocal operators

In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., $$\mathcal{L}_{p,s} = -Δ_p + (-Δ)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.

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