arXiv subjects

Daniele Cassani

Publications and source records attributed to Daniele Cassani.

At least 19 recordsLinked to original sources

arXiv2026

Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass

We study Hamiltonian elliptic systems with prescribed bilinear mass $\int_{\mathbb{R}^N}uv=a>0$. We establish a Gagliardo--Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve $1/p+1/q=N/(N+2)$ and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo--Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.

math.AP↗

arXiv2026

Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation

We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in $\mathbb R^N$, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on $\mathbb R^N\setminus\{0\}$. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to $u>0$ throughout $\mathbb R^N$, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.

math.AP↗

arXiv2026

Discrepancy geometry in approximate Bayesian inference: transport and risk perspectives

Approximate Bayesian Computation (ABC) replaces the evaluation of an intractable likelihood with comparisons between observed and simulated data. We develop a unified measure-theoretic framework for discrepancy-based Bayesian inference in which such comparisons are encoded through nonnegative compatibility weights. The proposed formulation provides a mathematical setting for studying both vanishing compatibility thresholds and large-sample asymptotic regimes, leading to convergence results and an asymptotic characterization of compatibility, including situations in which posterior concentration may fail. It also admits a natural variational interpretation through an entropy-regularized minimization principle and, when the discrepancy is induced by a Wasserstein distance, an intrinsic transport representation on spaces of probability measures, giving rise to risk-theoretic functionals. These results provide a unified probabilistic, asymptotic, variational, and geometric perspective on discrepancy-based Bayesian inference.

math.ST↗

arXiv2024

The mass-mixed case for normalized solutions to NLS equations in dimension two

\noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ .$$ We are interested in the so-called mass mixed case in which $f$ has $L^2$-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain pass type. We also investigate the asymptotic behavior of solutions approaching the zero mass case, namely when $c\to 0^+$.

math.AP↗

arXiv2024

Nonlocal Schrödinger-Poisson systems in $\mathbb R^N$: the fractional Sobolev limiting case

We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole $\mathbb R^N$. A quasilinear fractional Schrödinger equation, where the leading operator is the $\frac Ns$-fractional Laplacian, is coupled with a higher-order and possibly fractional Poisson equation. For both operators the dimension $N\geq 2$ corresponds to the limiting case of the Sobolev embedding, hence we consider nonlinearities with exponential growth. Since standard variational tools cannot be applied due to the sign changing logarithmic Riesz kernel of the Poisson equation, we employ a variational approximating procedure for an auxiliary Choquard equation, where the Riesz kernel is uniformly approximated by polynomial kernels. Qualitative properties of solutions such as symmetry, regularity and decay are also established. Our results extend and complete the analysis carried out in the planar case in [D. Cassani, Z. Liu, G. Romani. arxiv:2305.15274].

math.AP↗

arXiv2024

Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods

This paper deals with the following equation $$-Δu =K(|x'|, x'')\Big(|x|^{-α}\ast (K(|x'|, x'')|u|^{2^{\ast}_α})\Big) |u|^{2^{\ast}_α-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $α>5-\frac{6}{N-2}$, $2^{\ast}_α=\frac{2N-α}{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x'|, x'')$, where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.

math.AP↗

arXiv2023

Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case

We study the nonlinear Schrödinger equation for the $s-$fractional $p-$Laplacian strongly coupled with the Poisson equation in dimension two and with $p=\frac2s$, which is the limiting case for the embedding of the fractional Sobolev space $W^{s,p}(\mathbb{R}^2)$. We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.

math.AP↗

arXiv2023

Global vs blow-up solutions and optimal threshold for hyperbolic ODEs with possibly singular nonlinearities

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbation or large initial value, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold.

math.AP↗

arXiv2023

Positive solutions to the planar logarithmic Choquard equation via asymptotic approximation

In this paper we study the following nonlinear Choquard equation $$ -Δu+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$ vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space $H^1(\mathbb{R}^2)$. We give a new proof and at the same time extend part of the results established in [Cassani-Tarsi, Calc. Var. P.D.E. (2021)].

math.AP↗

arXiv2023

Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(Ω\right)\hookrightarrow L^{q}\left(Ω\right), \end{align*} where $N\geq1$, $0 2s$ and the so-called Sobolev limiting case $N=1$, $s=\frac{1}{2}$ and $p=2$, where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.

math.AP↗

arXiv2023

Let the paintings play

In this paper, we introduce a mathematical method to extract similarities between paintings and musical tracks. Our approach is based on the digitalization of both paintings and musical tracks by means of finite expansions in terms of orthogonal basis functions (with both Fourier and wavelet bases). The best fit between a specific painting and a sample of musical tracks from a given composer is achieved via an $L^2$ projection upon a finite-dimensional subspace. Several examples are provided for the analysis of a collection of works of art by the Italian artist Marcello Morandini. Finally, we have developed an original applet that implements the process above and which can be freely downloaded from the site https://github.com/pgerva/playing-paintings.git

cs.MM↗

arXiv2022

Quasilinear logarithmic Choquard equations with exponential growth in $\mathbb{R}^N$

We consider the $N$-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential $α$ is equal to the Euclidean dimension $N$, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space $W^{1,N}(\mathbb{R}^N)$, where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable $log$-weighted variant of the Pohozaev-Trudinger inequality which provides a proper functional framework where we use variational methods.

math.AP↗

arXiv2021

Blow-up phenomena and asymptotic profiles passing from $H^1$-critical to super-critical quasilinear Schrödinger equations

We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2Δu+V(x)u-\hbar^{2+γ}uΔu^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $γ\geq 0$ is a parameter with relevant physical interpretations, $V$ and $K$ are given potentials and $N\geq 5$. We investigate the concentrating behavior of solutions when $γ>0$ and, differently form the case $γ=0$ where the leading potential is $V$, the concentration is here localized by the source potential $K$. Moreover, surprisingly for $γ>0$ we find a different concentration behavior of solutions in the case $p=\frac{2N}{N-2}$ and when $\frac{2N}{N-2}<p<\frac{4N}{N-2}$. This phenomenon does not occur when $γ=0$.

math.AP↗

arXiv2021

Schrödinger-Newton equations in dimension two via a Pohozaev-Trudinger log-weighted inequality

We study the following Choquard type equation in the whole plane $(C) -Δu+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schrödinger potential and the nonlinearity $f(x,u)$, whose primitive in $u$ vanishing at zero is $F(x,u)$, exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev--Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to $(C)$.

math.AP↗

arXiv2021

Blow-up rate and local uniqueness for fractional Schrödinger equations with nearly critical growth

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation $ (-Δ)^s u+V(x)u= u^{2_s^*-1-\varepsilon} \ \ \text{in}\ \ \mathbb{R}^N, $ where $\varepsilon>0$, $s\in (0,1)$, $2^*_s:=\frac{2N}{N-2s}$, $N>4s$. We show that the ground state $u_{\varepsilon}$ blows up and precisely with the following rate $\|u_{\varepsilon}\|_{L^\infty(\mathbb{R}^N)}\sim \varepsilon^{-\frac{N-2s}{4s}}$, as $ε\rightarrow 0^+$. We also localize the concentration points and, in the case of radial potentials $V$, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.

math.AP↗

arXiv2020

Maximum Principle for Higher Order Operators in General Domains

We first prove De Giorgi type level estimates for functions in $W^{1,t}(Ω)$, $Ω\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi's classes as obtained in [Di Benedetto--Trudinger, AIHP (1984)] for functions in $W^{1,2}$. As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

math.AP↗

arXiv2019

Uniqueness results for higher order elliptic equations and systems

In this paper we develop a Gidas-Ni-Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet's: the obstruction is apparently a new phenomenon and seems due to some loss of information though far from being clear. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier's boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

math.AP↗

arXiv2018

Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems \begin{equation} \begin{cases} -Δu+λ_1u=μ_1u(e^{u^2}-1)+βv\left(e^{uv}-1\right) \text{ in } Ω, &\\ -Δv+λ_2v=μ_2v(e^{v^2}-1)+βu\left(e^{uv}-1\right)\text{ in } Ω, &\\ u,v\in H^1_0(Ω) \end{cases} \end{equation} where $Ω$ is a bounded smooth domain, $λ_1,λ_2>-Λ_1$ (the first eigenvalue of $(-Δ,H^1_0(Ω))$, $μ_1,μ_2>0$ and $β$ is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For `small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities.

math.AP↗