arXiv subjects

Eugenio Massa

Publications and source records attributed to Eugenio Massa.

6 recordsLinked to original sources

arXiv2022

A nonsmooth variational approach to semipositone quasilinear problems in $\mathbb{R}^N$

This paper concerns the existence of a solution for the following class of semipositone quasilinear problems \begin{equation*} \left \{ \begin{array}{rclcl} -Δ_p u = h(x)(f(u)-a),\ & u > 0 & \mbox{in} & \mathbb{R}^N, \end{array} \right. \end{equation*} where $1 0$, $ f:[0,+\infty) \to [0,+\infty)$ is a function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to (0,+\infty)$ is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for $a$ small enough. We also provide a version of Hopf's Lemma and a Liouville-type result for the $p$-Laplacian in the whole $\mathbb{R}^N$.

math.AP↗

arXiv2019

A family of entire functions connecting the Bessel function $J_1$ and the Lambert $W$ function

Motivated by the problem of determining the values of $α>0$ for which $f_α(x)=e^α- (1+1/x)^{αx},\ x>0$ is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $φ_α$, $α>0$, of entire functions such that $f_α(x) =\int_0^\infty e^{-sx}φ_α(s)\,ds, \ x>0.$ We show that each function $φ_α$ has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions $φ_α$, which turn out to be related to the well known Bessel function $J_1$ and the Lambert $W$ function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of $φ_α$ as $α$ increases from $0$ to $\infty$ and to obtain a very precise approximation of the largest $α>0$ such that $φ_α(s)\geq0,\, s>0$, or equivalently, such that $f_α$ is completely monotonic.

math.CA↗

arXiv2019

Sobolev versus Hölder local minimizers in degenerate Kirchhoff type problems

In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering several different interactions between the nonlocal term and the nonlinearity.

math.AP↗

arXiv2018

Characterization of Strict Positive Definiteness on products of complex spheres

In this paper we consider Positive Definite functions on products $Ω_{2q}\timesΩ_{2p}$ of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which $p$ and/or $q$ can be $1$ or $\infty$. The condition we obtain states that a suitable set in $\mathbb{Z}^2$, containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.

math.CA↗

arXiv2017

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.

math.CA↗

arXiv2017

On necessary conditions for the Comparison Principle and the Sub and Supersolutions Method for the stationary Kirchhoff Equation

In this paper we propose a counterexample to the validity of the Comparison Principle and of the Sub and Supersolution Method for nonlocal problems like the stationary Kirchhoff Equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term $M(\|u\|^2)$ is somewhere increasing with respect to the $H_0^1$-norm of the solution. Comparing with existing results, this fills a gap between known conditions on $M$ that guarantee or prevent these properties, and leads to a condition which is necessary and sufficient for the validity of the Comparison Principle.

math.AP↗