arXiv subjects

Alessandro Cannone

Publications and source records attributed to Alessandro Cannone.

4 recordsLinked to original sources

arXiv2026

Ground state solutions for Hartree type equations driven by superposition operators and Pohozaev Identity

We investigate Hartree-type equations driven by a nonlocal operator $\mathcal{L}_\mu$, defined as a superposition of fractional Laplacians through a signed Borel measure $\mu$. Under Berestycki-Lions type assumptions, we prove the existence of a Mountain Pass solution and show that its energy level coincides with the minimum on the Pohozaev manifold. We also establish the boundedness of non-negative solutions. The proof of this fact requires a careful use of the Sobolev embedding in the iterative argument and a delicate treatment of the integrals involved in the estimates, as well as a Kato-type inequality in our general setting. Finally, we establish a general Pohozaev identity for solutions under a suitable summability assumption.

math.AP↗

arXiv2025

Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -\Delta u=(|x|^{-(n-2)}\ast u^{p-\epsilon})u^{p-1-\epsilon}\quad \mbox{in}~~\Omega,~~ u=0\quad \mbox{on}~~\partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $\epsilon>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(\lambda_{i,\epsilon}, v_{i,\epsilon})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.

math.AP↗

arXiv2025

Polyharmonic Nonlinear Scalar Field Equations

In this paper, we present a result on the existence of ground state solutions for the polyharmonic nonlinear equation $(-\Delta)^m u=g(u)$, assuming that $g$ has a general subcritical growth at infinity, inspired by Berestycki and Lions \cite{BerestyckiLions}. In comparison with the biharmonic case studied in \cite{Med-Siem}, the presence of a higher-order operator gives rise to several analytical challenges, which are overcome in the present work. Furthermore, we establish a new polyharmonic logarithmic Sobolev inequality.

math.AP↗

arXiv2024

A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N

In the paper we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of estremal functions or blow-up, where the domain is the ball or the entire space. We also show that the extremal functions satisfy suitable Euler-Lagrange equations. When the domain is the entire space, such equation can be derived by N-Laplacian Schrodinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends [16] to any dimension N bigger than 2.

math.AP↗