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 Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

2020 ◽  
Author(s):  
Weiwei Ao ◽  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Mean Field ◽  
Liouville Type ◽  
Subcritical Case ◽  
Toda Systems ◽  
The Mean ◽  
Trudinger Inequality ◽  
Mean Field Equation

Abstract We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

scholarly journals LYAPUNOV FUNCTIONAL AND BLOW-UP RESULTS FOR A CLASS OF PERTURBATIONS OF SEMILINEAR WAVE EQUATIONS IN THE CRITICAL CASE

2012 ◽  
Vol 09 (02) ◽  
pp. 195-221 ◽  
Author(s):  
MOHAMED ALI HAMZA ◽  
HATEM ZAAG
Keyword(s):  
Wave Equation ◽  
Singular Solution ◽  
Lyapunov Functional ◽  
Wave Equations ◽  
Critical Case ◽  
Critical Power ◽  
Blow Up ◽  
Semilinear Wave Equation ◽  
Nirenberg Inequality ◽  
Blow Up Rate

We study a class of perturbations for the semilinear wave equation with critical power nonlinearity (in the conformal transform sense). Working in the framework of similarity variables, we introduce a Lyapunov functional for this problem. Using a two-step argument based on interpolation and a critical Gagliardo–Nirenberg inequality, we establish that the blow-up rate of any singular solution is given by the solution of the nonperturbed associated ODE, specifically u″ = up.

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.

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.

Improved Moser-Trudinger Inequality Involving Lp Norm in n Dimensions

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Jiuyi Zhu
Keyword(s):  
Blow Up ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Lp Norm ◽  
Blow Up Analysis ◽  
Trudinger Inequality

AbstractThe paper is concerned about an improvement of Moser-Trudinger inequality involving Lbe the first eigenvalue associated with n-Laplacian. We obtain the following strengthened Moser-Trudinger inequality with blow-up analysisfor 0 ≤ α < λ̅(Ω) and 1 < p ≤ n, and the supremum is infinity for α ≥ λ̅(Ω), where

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

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.

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