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Giuseppe Riey

Publications and source records attributed to Giuseppe Riey.

8 recordsLinked to original sources
arXiv2025

Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient

We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω \\ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} \] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \]

math.AP↗
arXiv2024

Concave solutions to Finsler $p$-Laplace type equations

We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions describing the anisotropies to be not even.

math.AP↗
arXiv2023

Normalised solutions and limit profiles of the defocusing Gross-Pitaevskii-Poisson equation

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-Δu+λu+|u|^2u =(I_α*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb R^3}u^2dx=ρ^2,$$ where $ρ^2>0$ is the prescribed mass of the solutions, $λ\in\mathbb R$ is an a-priori unknown Lagrange multiplier, and $I_α(x)=A_α|x|^{3-α}$ is the Riesz potential of order $α\in(0,3)$. When $α=2$ this problem appears in the models of self-gravitating Bose-Einstein condensates, which were proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We establish the existence of branches of normalised solutions to the GPP equation, paying special attention to the shape of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. The main novelty in this work is in the derivation of sharp asymptotic estimates on the mass-energy curves. These estimates allow us to show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation or to a compactly supported radial ground state of the integral Thomas-Fermi equation. The behaviour of normalized solutions depends sensitively on whether $α$ is greater than, equal to, or less than one.

math.AP↗
arXiv2021

Normalized solutions to mass supercritical Schrodinger equations with negative potential

We study the existence of positive solutions with prescribed $L^2$-norm for the Schrödinger equation \[ -Δu-V(x)u+λu=|u|^{p-2}u\qquadλ\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in\left(2+\frac 4 N,2^*\right)$, $2^*:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^*:=+\infty$ if $N=1,2$. We treat two cases. Firstly, under an explicit smallness assumption on $V$ and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on $V$, and that $V$ is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.

math.AP↗
arXiv2019

Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations

We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.

math.AP↗
arXiv2016

A regularity result for the p-laplacian near uniform ellipticity

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently close to $p_0=2$. We show that this phenomenon is driven by the classical Calderón-Zygmund constant. As a byproduct of our analysis we show that $C^{1,α}$ regularity improves up to $C^{1,1^-}$, when p is close enough to 2. This result we believe it is particularly interesting in higher dimensions $n>2,$ when optimal $C^{1,α}$ regularity is related to the optimal regularity of $p$-harmonic mappings, which is still open.

math.AP↗
arXiv2013

Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces

We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.

math.AP↗