Equivalent Moser type inequalities in R2 and the zero mass case

By Daniele Cassani, Federica Sani, Cristina Tarsi. 2014

84 citations recorded by the source corpus.

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space $\mathcal{D}^{1,2}(\mathbb{R}^2)$, completion of smooth compactly supported functions with respect to the Dirichlet norm $\|\nabla\cdot\|_2$, and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in $H^1(\mathbb{R}^2)$ such as the Adachi-Tanaka inequality and a version of Ruf's inequality.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2e20b1a0c53b7858ff249c43d917c4feb46e18a7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities (2015, methodology): “5) in dimension two in the nonsingular case β = 0 has also been given in [4] using the sharp critical Trudinger-Moser inequality in R(2).”
Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in $\R^2$ (2016, background): “y was introduced by Adimurthi and Yadava [3], see also de Figueiredo, Miyagaki and Ruf [18]. The ﬁrst version of the Pohozaev-Trundiger-Moser inequality in R2 was established by Cao in [12], see also [41, 2, 13], and reads as follows Trudinger-Moser Lemma 1.2. If α > 0 and u ∈ H1(R2), then Z R2 h”