Trudinger–Moser inequality on the whole plane and extremal functions

2016 ◽  
Vol 18 (05) ◽  
pp. 1550054 ◽  
Author(s):  
João Marcos do Ó ◽  
Manassés de Souza
Keyword(s):  
Sobolev Space ◽  
Blow Up ◽  
Extremal Functions ◽  
Text Setting ◽  
Compactness Result ◽  
Blow Up Analysis ◽  
Elliptic Estimates

In this paper, we study a class of Trudinger–Moser inequality in the Sobolev space [Formula: see text]. Setting [Formula: see text] we prove: [Formula: see text] [Formula: see text] for [Formula: see text] for [Formula: see text], and [Formula: see text] there exist extremal functions for [Formula: see text] if [Formula: see text]. Blow-up analysis, elliptic estimates and a version of compactness result due to Lions are used to prove (1) and (3). The proof of (2) is based on computations of testing functions which are a combination of eigenfunctions with the Moser sequence.

Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)

2019 ◽  
Vol 21 (04) ◽  
pp. 1850023 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Critical Case ◽  
Blow Up ◽  
Sharp Inequality ◽  
Extremal Functions ◽  
Analysis Method ◽  
Subcritical Case ◽  
Best Constant ◽  
Blow Up Analysis ◽  
Nirenberg Inequality ◽  
Trudinger Inequality

We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

scholarly journals COMPACTNESS AND EXISTENCE RESULTS FOR DEGENERATE CRITICAL ELLIPTIC EQUATIONS

2005 ◽  
Vol 07 (01) ◽  
pp. 37-73 ◽  
Author(s):  
VERONICA FELLI ◽  
MATTHIAS SCHNEIDER
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
A Priori Estimate ◽  
Continuous Functions ◽  
Homotopy Invariance ◽  
Existence Of A Solution ◽  
Compactness Result ◽  
Analysis Techniques ◽  
Blow Up Analysis

This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli–Kohn–Nirenberg type. By means of blow-up analysis techniques, we prove an a priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray–Schauder degree.

scholarly journals Improved Adams-type inequalities and their extremals in dimension 2m

2020 ◽  
pp. 2050043
Author(s):  
Azahara DelaTorre ◽  
Gabriele Mancini
Keyword(s):  
Sobolev Space ◽  
Open Subset ◽  
Extremal Function ◽  
Blow Up ◽  
New Techniques ◽  
Blow Up Analysis ◽  
Bounded Smooth ◽  
Trudinger Inequality

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

scholarly journals Blow-Up Analysis for the Periodic Two-Component μ-Hunter-Saxton System

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yunxi Guo ◽  
Tingjian Xiong
Keyword(s):  
Transport Equation ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Blow Up Analysis ◽  
Classic Method ◽  
Two Component ◽  
Localization Analysis ◽  
Spatially Periodic ◽  
Periodic Setting

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.

Blow-Up Analysis for Heat Flow of Harmonic Maps

Nematics ◽  
1991 ◽  
pp. 49-64
Author(s):  
Chen Yunmei ◽  
Ding Wei-Yue
Keyword(s):  
Heat Flow ◽  
Blow Up ◽  
Harmonic Maps ◽  
Blow Up Analysis
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