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Natalino Borgia

Publications and source records attributed to Natalino Borgia.

6 recordsLinked to original sources
arXiv2026

Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-ε})u^{2_{s}^{\sharp}-2-ε} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}Ω,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}Ω,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\partialΩ,\end{aligned} \right.\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$, $0 0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the spectral fractional Laplacian. For a general domain $Ω$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $ε\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

math.AP↗
arXiv2026

Variational and Geometric Analysis for Quasilinear Elliptic Equations and Systems

In this thesis we focus on quasilinear elliptic systems driven by various nonlinear operators, such as the p-Laplacian, and nonlinear sources that are allowed to exhibit both subcritical and critical growth. We aim to establish the existence of solutions for perturbation of specific eigenvalue problems, by employing variational and topological methods. To establish existence results for autonomous systems of quasilinear PDEs in the spirit of the paper by Amann and Zehnder, we develop a local Morse theory for functional associated to quasilinear elliptic systems. By refining topological arguments introduced by Cingolani and Degiovanni in Banach product spaces, we establish the finiteness of the critical groups and we derive a Poincaré-Hopf formula in a Banach product space, in presence of both subcritical and critical nonlinear coupling. We also establish uniform boundedness results for anisotropic quasilinear systems, that are of interest within regularity theory. To show existence results for non-autonomous systems of quasilinear PDEs in the spirit of the paper of Landesman and Lazer, we consider the eigenvalue problem for quasilinear elliptic systems introduced by de Thélin. We prove the simplicity and isolation of the first eigenvalue lambda1. Furthermore, we show the existence of a sequence of eigenvalues by employing a suitable deformation lemma proved by Bonnet. Subsequently, we analyze new sufficient Landesman-Lazer type conditions within the framework of quasilinear elliptic systems. We also investigate the N-dimensional Euclidean Onofri inequality, established by Del Pino and Dolbeault for smooth functions with compact support. After extending this inequality to a suitable weighted Sobolev space, we exploit its connection with the Liouville equation on R^N to prove an equivalence with the sharp logarithmic Moser-Trudinger inequality on the unit ball of R^N.

math.AP↗
arXiv2026

On the de Thélin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

In this paper we prove that the smallest eigenvalue $λ_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{λ_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $λ_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

math.AP↗
arXiv2025

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 $\mathbb{R}^N$.

math.AP↗
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.

math.AP↗