arXiv subjects

Soledad Benguria

Publications and source records attributed to Soledad Benguria.

5 recordsLinked to original sources

arXiv2019

The Brezis-Nirenberg problem for the Laplacian with a singular drift in $\mathbb{R}^n$ and $\mathbb{S}^n.$

We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, $δ$, and $λ$, the coupling constant of the linear term of the Brezis-Nirenberg problem.

math.AP↗

arXiv2019

A Non-Existence Result for a Generalized Radial Brezis-Nirenberg Problem

We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.

math.AP↗

arXiv2016

An application of John ellipsoids to the Szego kernel on unbounded convex domains

We use convex geometry tools, in particular John ellipsoids, to obtain a size estimate for the Szegő kernel on the boundary of a class of unbounded convex domains in $\mathbb{C}^n.$ Given a polynomial $b:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying a certain growth condition, we consider domains of the type $Ω_b = \{ z\in\mathbb{C}^{n+1}\,:\, {\rm Im}[z_{n+1}] > b({\rm Re}[z_1],\ldots,{\rm Re}[z_n]) \}.$

math.CV↗

arXiv2016

An improved bound for the non-existence of radial solutions of the Brezis-Nirenberg problem in Hn

Using a Rellich-Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis-Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of Hn, for 2<n<4.

math.AP↗

arXiv2016

The solution gap of the Brezis-Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem $-Δ_{\mathbb{H}^n} u = λu + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(Ω),$ where $Ω$ is a geodesic ball of radius $θ_1$ on $\mathbb{H}^n.$ For radial solutions, this equation can be written as an ODE having $n$ as a parameter. In this setting, the problem can be extended to consider real values of $n.$ We show that if $2 0$ if $0 < θ<θ_1$ and $P_{\ell}^{-α}(\coshθ_1)=0,$ with $α= (2-n)/2.$

math.AP↗