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.

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.].

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.

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

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.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Multiple boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

Sign in / Sign up

Export Citation Format

Share Document