On local and nonlocal Moser-Trudinger inequalities

Moser-Trudinger inequalities arise naturally in the study of the critical case of the well known Sobolev embeddings. In this work, we cover two issues related to Moser-Trudinger inequalities. We address the problem of the existence of extremal functions for Moser-Trudinger embeddings in the presence of singular potentials, and the problem of finding sharp Moser-Trudinger type inequalities in fractional Sobolev spaces

A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one

We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole $\mathbb{R}$ and we give an answer to one of their open questions

A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one

We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole $\mathbb{R}$ and we give an answer to one of their open questions

Extremal functions for singular Moser-Trudinger embeddings

We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri’s inequality for the unit disk in $\mathbb{R}^2$. Moreover we extend the analysis of [1] and [8] considering Adimurthi-Druet type functionals on compact surfaces with conical singularities and discussing the existence of extremals for such functionals

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^{\infty}(\Omega)$ such that $\Delta^m \varphi \equiv 0$ and bounded sequence $(V_k) \subset L^\infty (\Omega)$ we prove the existence of a sequence of functions $(u_k) \subset C^{2m-1}(\Omega)$ solving the Liouville equation of order 2m $$(-\Delta)^m u_k = V_k e^{2m u_k} in \Omega, \limsup_{k\to\infty} \int_{\Omega} e^{2m u_k} dx < \infty,$$ and blowing up exactly on the set $S_{\varphi} := \{x\in\Omega : \varphi(x) = 0\}$, i.e. $$\lim_{k\to\infty} u_k(x) = +\infty for x\in S_{\varphi} and \lim_{k\to\infty} u_k(x) = -\infty for x\in\Omega\setminus S_{\varphi},$$ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $\Omega$ and to the case $\Omega = \mathbb{R}^{2m}$. Several related problems remain open

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^{\infty}(\Omega)$ such that $\Delta^m \varphi \equiv 0$ and bounded sequence $(V_k) \subset L^\infty (\Omega)$ we prove the existence of a sequence of functions $(u_k) \subset C^{2m-1}(\Omega)$ solving the Liouville equation of order 2m $$(-\Delta)^m u_k = V_k e^{2m u_k} in \Omega, \limsup_{k\to\infty} \int_{\Omega} e^{2m u_k} dx < \infty,$$ and blowing up exactly on the set $S_{\varphi} := \{x\in\Omega : \varphi(x) = 0\}$, i.e. $$\lim_{k\to\infty} u_k(x) = +\infty for x\in S_{\varphi} and \lim_{k\to\infty} u_k(x) = -\infty for x\in\Omega\setminus S_{\varphi},$$ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $\Omega$ and to the case $\Omega = \mathbb{R}^{2m}$. Several related problems remain open

Current trends in the real-life use of dalbavancin: report of a study panel.

Dalbavancin is a novel lipoglycopeptide antibiotic with a chemical structure similar to teicoplanin. Dalbavancin has been approved and marketed since 2014 in the USA and 2015 in the European Union for the treatment of acute bacterial skin and skin-structure infections (ABSSSIs) caused by Gram-positive cocci. ABSSSIs include infectious syndromes such as erysipelas, cellulitis, major cutaneous abscesses that require incision and drainage, and both surgical and traumatic wound infections. In current clinical practice, dalbavancin is also used for cardiac implantable electronic device-related soft tissue infection and other prosthetic infections, and therefore when the presence of biofilm is a concern. In this review, we aimed to highlight our experience with the use of dalbavancin for some of the most hard-to-treat Gram-positive infections, as well as a promising strategy in terms of pharmacoeconomic effectiveness. We describe our current real-life clinical practice with the use of dalbavancin, depicting a few representative clinical cases in order to share our own practice in the hospital setting

Large blow-up sets for the prescribed Q -curvature equation in the Euclidean space

Let m≥2 be an integer. For any open domain Ω⊂R2m, non-positive function φ∈C∞(Ω) such that Δmφ≡0, and bounded sequence (Vk)⊂L∞(Ω) we prove the existence of a sequence of functions (uk)⊂C2m−1(Ω) solving the Liouville equation of order 2m (−Δ)muk=Vke2mukin Ω,limsupk→∞∫Ωe2mukdx<∞, and blowing up exactly on the set Sφ:={x∈Ω:φ(x)=0}, i.e. limk→∞uk(x)=+∞ for x∈Sφandlimk→∞uk(x)=−∞ for x∈Ω∖Sφ, thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of Ω and to the case Ω=R2m. Several related problems remain open

Adenovirus: an overview for pediatric infectious diseases specialists

We show a sharp fractional Moser-Trudinger type inequality in dimension 1, i.e. for an interval $I \Subset \mathbb{R}, p \in (1,\infty)$ and some $\alpha > 0$ $$\sup_{u\in\tilde{H}^{{1/p},p}(I):\|(-\Delta)^{1/2p} u\|_{L^p(I)} \le 1} \int_I |u|^a e^{\alpha_p |u|^{p/(p-1)} dx} < \infty \text{ if and only if } a = 0.$$ Here $\tilde{H}^{{1/p},p}(I) = \{u \in L^p(\mathbb{R}) : (-\Delta)^{1/2p} u \in L^p(\mathbb{R}), \operatorname{supp}(u)\subset I\}$. Restricting ourselves to the case p = 2 we further consider for $\lambda > 0$ the functional $$J(u) := 1/2 \int_{\mathbb{R}} |(-\Delta)^{1/4} u|^{2} dx - \lambda \int_I (e^{1/2 u^2} - 1) dx, u\in \tilde{H}^{{1/2},2}(I),$$ and prove that it satisfies the Palais-Smale condition at any lever $c \in (-\infty, \pi)$. We use these results to show that the equation $$(-\Delta)^{1/2} u = \lambda u e^{1/2 u^2} \text{ in } I$$ has a positive solution in $\tilde{H}^{{1/2},2}(I)$ if and only if $\lambda \in (0, \lambda_1 (I) )$, where $\lambda_1$ is the first eigenvalue of $(-\Delta)^{n/2}$ on I. This extends to the fractional case some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators. Finally with a technique of Ruf we show a fractional Moser-Trudinger inequality on $\mathbb{R}$