arXiv subjects

Leandro Milne

Publications and source records attributed to Leandro Milne.

3 recordsLinked to original sources

arXiv2024

The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series

Given a frequency $\lambda=(\lambda_n)$, we consider the Hardy spaces $ \mathcal{H}_p^\lambda$ of $\lambda$-Dirichlet series $ D = \sum_n a_n e^{-\lambda_n s}$ and study the asymptotic behavior of the upper and lower democracy functions of its canonical basis $\mathcal B=\{e^{-\lambda_ns}\}$. For the ordinary case, $\mathcal B=\{n^{-s}\}$, we give the correct asymptotic behavior of all such functions, while in the general case we give sharp lower and upper bounds for all possible behaviors. Moreover, for $p>2$ we present examples showing that any intermediate behavior (between the extreme bounds) can occur. We also study how different properties of the frequency $\lambda$ lead to particular behaviors of the corresponding fundamental functions. Finally, we apply our results to analyze greedy-type properties of $\mathcal B=\{e^{-\lambda_ns}\}$ for some particular $\lambda$'s.

math.FA↗

arXiv2023

Positive solutions for concave-convex type problems for the one-dimensional $\phi$-Laplacian

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -\phi\left( u^{\prime}\right) ^{\prime}=\lambda m(x)f(u)+\mu n(x)g(u) & \text{ in }\Omega,\\ u=0 & \text{ on }\partial\Omega, \end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\phi$, respectively. Our assumptions on $\phi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\u{\i}\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.

math.CA↗

arXiv2017

Positive solutions for nonlinear problems involving the one-dimensional ϕ-Laplacian

Let $Ω:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( Ω\right) $ and $λ>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-ϕ\left( u^{\prime}\right) ^{\prime}+r\left( x\right) ϕ\left( u\right) $, where $ϕ:\mathbb{R\rightarrow R}$ is an odd increasing homeomorphism and $0\leq r\in L^{1}\left( Ω\right) $. We study the existence of positive solutions for problems of the form $\mathcal{L}u=λm\left( x\right) f\left( u\right)$ in $Ω,$ $u=0$ on $\partialΩ$, where $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is a continuos function which is, roughly speaking, sublinear with respect to $ϕ$. Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases $r\equiv0$ and/or $m\geq0$.

math.CA↗