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2. Extremals of singular Hardy–Trudinger–Moser inequality with remainder terms on unit disc
- Author
-
Wang, Weiwei
- Published
- 2024
- Full Text
- View/download PDF
3. Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation.
- Author
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Ben Ayed, Mohamed, El Mehdi, Khalil, and Mohamed Salem, Fatimetou
- Subjects
NONLINEAR equations ,NEUMANN problem ,VECTOR fields ,PARTIAL differential equations ,SMOOTHNESS of functions - Abstract
In this paper, we consider the nonlinear Neumann problem (P ε) : − Δ u + V (x) u = K (x) u (n + 2) / (n − 2) − ε , u > 0 in Ω , ∂ u / ∂ ν = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n ≥ 4 , ε is a small positive real, and V and K are non-constant smooth positive functions on Ω ¯ . First, we study the asymptotic behavior of solutions for (P ε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K ≡ 1 , problem (P ε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (P ε) which allow us to provide multiplicity results for (P ε) . The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Extremal functions for a singular super-critical Trudinger-Moser inequality.
- Author
-
Juan Zhao
- Subjects
SYMMETRIC functions ,SURFACE area ,EIGENVALUES - Abstract
In this paper, we deal with a singular super-critical Trudinger-Moser inequality on a unit ball of R
n , n ≥ 3. For any p > 1, we set λp (B) = infu∈W ∫0 1,n (B),u̸≡0B |∇u|n dx/ ( ∫B |u|p dx)n/p as an eigenvalue related to the n-Laplacian. Let S be a set of radially symmetric functions. Precisely, if β ≥ 0 and α < (1 + p/n β)n−1+n/pλ (B), then there exists a positive constant ϵ0 such that when λ ≤ 1 + ϵp 0 ,supu∈W ∫0 1,n (B)∩S, ∫B |∇u|n dx−α( ∫ B |u|p|x|pβ dx)n/p ≤1B |x| pβ (e αn (1+ p/n β)|u|n/n−1 − λ ∑k=0 m |α . The proof is based on the method of blow-up analysis. The case λ = 0 was studied by Yang-Zhu in [38]. de Figueiredo [11] considered the case p = 2, β ≥ 0, and α = 0 in two dimension. The case λ = 0, p = n, −1 < β < 0, and α = 0 was considered by Adimurthi-Sandeep [1]. Our results extend those of the above cases. [ABSTRACT FROM AUTHOR]n (1 + p/n β)un/n−1 |k1/(n−1) n−1 , ωn−1 is the surface area of the unit ball in Rn . The proof is based on the method of blow-up analysis. The case λ = 0 was studied by Yang-Zhu in [38]. de Figueiredo [11] considered the case p = 2, β ≥ 0, and α = 0 in two dimension. The case λ = 0, p = n, −1 < β < 0, and α = 0 was considered by Adimurthi-Sandeep [1]. Our results extend those of the above cases. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
5. A sharp Moser-Trudinger type inequality involving Lp norm in [formula omitted] with degenerate potential.
- Author
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Sun, Jingxuan, Song, Zhen, and Zou, Wenming
- Subjects
- *
SOBOLEV spaces , *EIGENVALUES - Abstract
In this paper, we establish a Moser-Trudinger type inequality with a special degenerate potential V. Specifically, we demonstrate that for any potentials V (x) satisfying the conditions (V 1) and (V 2) , the following inequality MT (V , α , p) ≔ sup u ∈ E , ‖ u ‖ ≤ 1 ∫ R n Φ (α n (1 + α ‖ u ‖ p n) 1 n − 1 u n n − 1 ) d x < ∞ holds if α < λ 1 (V) ; where E is the usual weighted Sobolev space associated to the potential V , the norm ‖ u ‖ n : = ∫ R n | ∇ u | n + V (x) | u | n d x , and λ 1 (V) is the first Dirichlet eigenvalue of the corresponding operator − Δ + V. The potential conditions are as follows. (V 1) 0 ≤ m = inf x ∈ R n V (x) < sup x ∈ R n V (x) = lim | x | → ∞ V (x) = M ≤ ∞. (V 2) V is radially symmetric, non-decreasing. Besides, we also prove that for any potentials V (x) satisfying the condition (V 1) merely and for p = n , the previous inequality is still valid. In addition, this inequality is sharp in the sense that if p > n and α ≥ λ 1 (V) , there holds MT (V , α , p) = ∞. Meanwhile, if α > λ 1 (V) , then MT (V , α , n) = ∞. Furthermore, via a subtle blow-up analysis, we also prove the existence of an extremal function for the inequality above when α is sufficiently small. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. A nonexistence result for blowing up sign-changing solutions of the Brezis-Nirenberg-type problem.
- Author
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DAMMAK, Yessine
- Subjects
BOUNDED arithmetics ,SOBOLEV spaces ,COMPACT spaces (Topology) ,TOPOLOGICAL spaces ,TOPOLOGY - Abstract
We consider the Brezis-Nirenberg problem: -Δu = |u|
p-1 u ± εu in Ω, with u = 0 on ∂Ω, where Ω is a smooth bounded domain in Rn , n ≤ 4, p + 1 = 2n/(n - 2) is the critical Sobolev exponent, and ε > 0 is a positive parameter. The main result of this paper shows that if n ≤ 4 there are no sign-changing solutions uε of (P-ε ) with two positive and one negative blow up points. [ABSTRACT FROM AUTHOR]- Published
- 2018
- Full Text
- View/download PDF
7. THE NONEXISTENCE OF EXTREMALS FOR THE HARDY-TRUDINGER-MOSER INEQUALITY IN THE HYPERBOLIC SPACE.
- Author
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QIANJIN LUO
- Subjects
HYPERBOLIC spaces ,FUNCTION spaces ,MATHEMATICS theorems ,MATHEMATICAL formulas ,INTEGERS - Abstract
Let B be the unit disc in R², H be the completion of C
∞ 0 (B) under the norm ... We prove that the supremum in the following inequality ... can not be achieved by any functions in the function space H when α is sufficiently close to λ- 1 , i.e., 0 < λ1 - α ... 1, where ... Evidently, this conclusion is complementary to that of [12, Theorem 1.1 (ii)]. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
8. A SHARP TRUDINGER-MOSER TYPE INEQUALITY IN ℝ².
- Author
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DE SOUZA, MANASSÉS and MARCOS DO Ó, JOÃO
- Subjects
SCHRODINGER operator ,ORLICZ spaces ,SOBOLEV spaces ,BANACH spaces ,ELLIPTIC equations ,LEBESGUE measure - Abstract
In this paper, we establish a sharp Trudinger-Moser type inequality for a class of Schrödinger operators in ℝ². We obtain a result related to the compactness of the embedding of a subspace of W
1,2 (ℝ²) into the Orlicz space Lf (ℝ²) determined by f(t) = eßt² -1. Our result is similar to one obtained by Adimurthi and Druet for smooth bounded domains in ℝ², which is closely related to a compactness result proved by Lions. Furthermore, similarly to what has been done by Carleson and Chang, we prove the existence of an extremal function for this Trudinger-Moser inequality by performing a blow-up analysis. Trudinger-Moser type inequalities have a wide variety of applications to the study of nonlinear elliptic partial differential equations involving the limiting case of Sobolev inequalities and have received considerable attention in recent years. [ABSTRACT FROM AUTHOR]- Published
- 2014
- Full Text
- View/download PDF
9. Anisotropic Moser-Trudinger inequality involving Ln norm.
- Author
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Zhou, Changliang
- Subjects
- *
MATHEMATICAL equivalence , *UNIT ball (Mathematics) , *BLOWING up (Algebraic geometry) , *FINSLER spaces - Abstract
The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves L n norm. Let λ 1 (Ω) = inf u ∈ W 0 1 , n (Ω) , u ≢ 0 | | F (∇ u) | | L n (Ω) n / | | u | | L n (Ω) n be the first eigenvalue associated with n -Finsler-Laplacian. Using blow-up analysis, we obtain that sup u ∈ W 0 1 , n (Ω) , | | F (∇ u) | | L n (Ω) = 1 ∫ Ω e λ n (1 + α | | u | | L n (Ω) n) 1 n − 1 | u | n n − 1 d x is finite for any 0 ≤ α < λ 1 (Ω) , and the supremum is infinite for any α ≥ λ 1 (Ω) , where λ n = n n n − 1 κ n 1 n − 1 (κ n is the volume of the unit wulff ball) and the function F is positive, convex and homogeneous of degree 1, and its polar F o represents a Finsler metric on R n. Furthermore, the supremum is attained for any 0 ≤ α < λ 1 (Ω). [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
10. Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation
- Author
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Mohamed Ben Ayed, Khalil El Mehdi, and Fatimetou Mohamed Salem
- Subjects
partial differential equations ,Schrödinger equation ,Neumann elliptic problems ,blow-up analysis ,critical Sobolev exponent ,Mathematics ,QA1-939 - Abstract
In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.
- Published
- 2024
- Full Text
- View/download PDF
11. Regularity condition of the incompressible Navier–Stokes equations in terms of one velocity component.
- Author
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Bae, Hantaek and Kang, Kyungkeun
- Subjects
- *
NAVIER-Stokes equations , *VELOCITY - Abstract
Abstract In this paper, we provide a new regularity criterion of smooth solutions via the one component of the velocity field in various scaling invariant spaces with a natural growth condition of the L ∞ norm near a possible blow-up time. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
12. Topological methods for the resonant [formula omitted]-curvature problem in arbitrary even dimension.
- Author
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Ndiaye, Cheikh Birahim
- Subjects
- *
RIEMANNIAN manifolds , *MORSE theory , *DIMENSIONS , *INTEGERS - Abstract
Abstract This paper is devoted to the problem of existence of conformal metrics with prescribed Q -curvature on closed Riemannian manifolds of even dimension n ≥ 4 , when the kernel of the associated GJMS operator is trivial and the total integral of the corresponding Q -curvature is a positive integer multiple of the one of the n -dimensional round sphere. Indeed, exploiting the variational structure of the problem, we develop a full Morse theory and algebraic topological arguments for existence for this non-compact geometric variational problem of high order, extending the works Ahmedou and Ndiaye (2019) and Ndiaye (2015) to arbitrary even dimensions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
13. Symmetry analysis, conserved quantities and applications to a dissipative DGH equation.
- Author
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Wei, Long and Wang, Yang
- Subjects
- *
SYMMETRY , *CONSERVATION laws (Mathematics) , *LIE series , *NONLINEAR analysis , *EQUATIONS - Abstract
Abstract In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
14. Some sharp inequalities related to Trudinger–Moser inequality.
- Author
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de Souza, Manassés
- Subjects
- *
MATHEMATICAL inequalities , *NONLINEAR operators , *DENSITY functional theory , *PROBLEM solving , *ALGORITHMS - Abstract
This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
15. Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space.
- Author
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Li, Xiaomeng and Yang, Yunyan
- Subjects
- *
EUCLIDEAN geometry , *EUCLIDEAN domains , *PRINCIPAL ideal domains , *GEOMETRIC congruences , *HILBERT'S problems - Abstract
In a previous work (Adimurthi and Yang, 2010 [2] ), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space R N ( N ≥ 2 ) . Precisely, if 0 ≤ β < 1 and 0 < γ ≤ 1 − β , then there holds for any τ > 0 , sup u ∈ W 1 , N ( R N ) , ∫ R N ( | ∇ u | N + τ | u | N ) d x ≤ 1 ∫ R N 1 | x | N β ( e α N γ | u | N N − 1 − ∑ k = 0 N − 2 α N k γ k | u | k N N − 1 k ! ) d x < ∞ , where α N = N ω N − 1 1 / ( N − 1 ) and ω N − 1 is the area of the unit sphere in R N . The above inequality is sharp in the sense that if γ > 1 − β , all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β = 0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11] . We shall investigate the singular case 0 < β < 1 and prove that for all τ > 0 , 0 < β < 1 and 0 < γ ≤ 1 − β , extremal functions for the above inequalities exist. The proof is based on blow-up analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
16. A non-compactness result on the fractional Yamabe problem in large dimensions.
- Author
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Kim, Seunghyeok, Musso, Monica, and Wei, Juncheng
- Subjects
- *
GEOMETRIC modeling , *FUNCTIONAL analysis , *VECTOR spaces , *TOPOLOGY , *MATHEMATICAL models - Abstract
Let ( X n + 1 , g + ) be an ( n + 1 ) -dimensional asymptotically hyperbolic manifold with conformal infinity ( M n , [ h ˆ ] ) . The fractional Yamabe problem addresses to solve P γ [ g + , h ˆ ] ( u ) = c u n + 2 γ n − 2 γ , u > 0 on M where c ∈ R and P γ [ g + , h ˆ ] is the fractional conformal Laplacian whose principal symbol is the Laplace–Beltrami operator ( − Δ ) γ on M . In this paper, we construct a metric on the half space X = R + n + 1 , which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that n ≥ 24 for γ ∈ ( 0 , γ ⁎ ) and n ≥ 25 for γ ∈ [ γ ⁎ , 1 ) where γ ⁎ ∈ ( 0 , 1 ) is a certain transition exponent. The value of γ ⁎ turns out to be approximately 0.940197. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
17. A non-existence result for low energy sign-changing solutions of the Brezis–Nirenberg problem in dimensions 4, 5 and 6.
- Author
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Dammak, Yessine
- Subjects
- *
EXISTENCE theorems , *CRITICAL exponents , *BLOWING up (Algebraic geometry) , *COMPACT spaces (Topology) , *SOBOLEV spaces - Abstract
We consider the Brezis–Niremberg problem: ( P ε ) { − △ u = | u | p − 1 u + ε u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n = 4 , 5 , 6 , p + 1 = 2 n n − 2 is the critical Sobolev exponent and ε is a positive parameter. The main result of the paper generalizes the result of A. Iacopetti and F. Pacella [10] . Precisely we show that there are no low energy sign-changing solutions u ε with max u ε / min u ε → 0 or −∞ as ε goes to zero. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
18. Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two.
- Author
-
Yang, Yunyan and Zhu, Xiaobao
- Subjects
- *
BLOWING up (Algebraic geometry) , *MATHEMATICAL inequalities , *FUNCTIONAL analysis , *MATHEMATICS theorems , *MATHEMATICAL analysis - Abstract
In this paper, we derive a sharp version of the singular Trudinger–Moser inequality, which was originally established by Adimurthi and Sandeep [2] . Moreover, extremal functions for those singular Trudinger–Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work ( [32] ), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behavior of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li [6,7] to get the desired bubble. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
19. Blow-up analysis for a cosmic strings equation.
- Author
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Tarantello, Gabriella
- Subjects
- *
STRING models (Physics) , *TOPOLOGICAL defects (Physics) , *COSMIC strings , *STRING cosmology , *METAPHYSICAL cosmology - Abstract
In this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the “uniformization” of the Riemann sphere with conical singularities. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
20. COMPACTNESS OF EXTREMALS FOR SINGULAR TRUDINGER--MOSER INEQUALITIES ON THE WHOLE EUCLIDEAN SPACE.
- Author
-
XIAOMENG LI, LIU YANG, and XIANFENG SU
- Subjects
EUCLIDEAN algorithm ,MATHEMATICAL variables ,MATHEMATICAL equivalence ,MATHEMATICS theorems ,MATHEMATICAL constants - Abstract
Let W
1,n (ℝn ) be the standard Sobolev space. For any τ > 0, 0 < β < 1, Li and Yang [16] proved the existence of extremals for a singular Trudinger-Moser inequality. Namely, the supremum ... can be attained by some function uβ ∈ W1,n (ℝn ) with ... (|∇u|n + τ|u|n )dx=1. Here Φ(n,t)=et ...!, and αn =nωn-1 1/(n-1) with ωn-1 being the surface area of the (n--1) -dimensional unit sphere. In this note, we consider the compactness of the function family {uβ }0<β<1 and prove that up to a subsequence, uβ converges to some function u0 in C¹(ℝn ) when β→ 0. Moreover, u0 is an extremal function of the supremum ... Let us explain the result in geometry. Denote ω0 (x) = ∑j=1 n dxj ² and ωβ (x) = |x|-2β ω0 (x) as the standard and conical metrics on ℝn . Then the extremal family {uβ }0<β<1 of the following singular Trudinger-Moser functionals ... is compactness as β→0. This extends earlier result of Wang and Yang [33] and complements that of Li and Yang [16]. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
21. Extremal functions for Trudinger–Moser inequalities with nonnegative weights
- Author
-
Hou, Songbo
- Published
- 2018
- Full Text
- View/download PDF
22. Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two.
- Author
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Yang, Yunyan
- Subjects
- *
MATHEMATICAL inequalities , *EXISTENCE theorems , *BLOWING up (Algebraic geometry) , *RIEMANNIAN geometry , *MATHEMATICAL functions - Abstract
Combining Carleson–Chang's result [9] with blow-up analysis, we prove existence of extremal functions for certain Trudinger–Moser inequalities in dimension two. This kind of inequality was originally proposed by Adimurthi and O. Druet [1] , extended by the author to high dimensional case and Riemannian surface case [40,41] , generalized by C. Tintarev to wider cases including singular form [36] and by M. de Souza and J.M. do Ó [14] to the whole Euclidean space R 2 . In addition to the Euclidean case, we also consider the Riemannian surface case. The results in the current paper complement that of L. Carleson and A. Chang [9] , M. Struwe [35] , M. Flucher [16] , K. Lin [19] , and Adimurthi and O. Druet [1] , our previous ones [26,41] , and part of C. Tintarev [36] . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
23. Blow-up analysis of a Finsler–Liouville equation in two dimensions
- Author
-
Wang, Guofang and Xia, Chao
- Subjects
- *
BLOWING up (Algebraic geometry) , *STURM-Liouville equation , *DIMENSIONAL analysis , *EXISTENCE theorems , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, we establish a blow-up analysis for the Finsler–Liouville equation in two dimensions. Here the operator Q is the so-called Finsler–Laplacian, which is a generalized Laplacian. As a byproduct, we obtain an existence result. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
24. Classification of low energy sign-changing solutions of an almost critical problem
- Author
-
Ben Ayed, Mohamed, El Mehdi, Khalil, and Pacella, Filomena
- Subjects
- *
FUNCTIONAL analysis , *CALCULUS of variations , *MAXIMA & minima , *MATHEMATICS - Abstract
Abstract: In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
25. Blow-up and symmetry of sign-changing solutions to some critical elliptic equations
- Author
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Ben Ayed, Mohamed, El Mehdi, Khalil, and Pacella, Filomena
- Subjects
- *
EQUATIONS , *MATHEMATICS , *MATHEMATICAL symmetry , *GROUP theory - Abstract
Abstract: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
26. Blow-up and nonexistence of sign changing solutions to the Brezis–Nirenberg problem in dimension three
- Author
-
Ben Ayed, Mohamed, El Mehdi, Khalil, and Pacella, Filomena
- Subjects
- *
EXPONENTS , *EXPONENTIAL functions , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: In this paper we study low energy sign changing solutions of the critical exponent problem in Ω, on ∂Ω, where Ω is a smooth bounded domain in and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
27. The prescribed boundary mean curvature problem on <f>B4</f>
- Author
-
Djadli, Zindine, Malchiodi, Andrea, and Ahmedou, Mohameden Ould
- Subjects
- *
CURVATURE , *CURVES , *CALCULUS , *NUMERICAL analysis - Abstract
In this paper, we perform a fine blow-up analysis for a boundary value elliptic equation involving the critical trace Sobolev exponent related to the conformal deformation of the metrics on the standard ball, namely the problem of prescribing the boundary mean curvature. From this analysis some a priori estimates in low dimension are obtained. With these estimates, we prove the existence of at least one solution when an index-counting formula associated to the prescribed mean curvature is different from zero. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
28. A compactness theorem for the fractional Yamabe problem, Part I: The nonumbilic conformal infinity.
- Author
-
Seunghyeok Kim, Musso, Monica, and Juncheng Wei
- Subjects
COMPACT spaces (Topology) ,MANIFOLDS (Mathematics) ,STOCHASTIC convergence ,CURVATURE ,PROOF theory - Abstract
Assume that (X,g
+ ) is an asymptotically hyperbolic manifold, (M,[h¯ ]) is its conformal infinity, ρ is the geodesic boundary defining function associated to h¯ and g¯=ρ²g+ . For any γ∈(0,1), we prove that the solution set of the γ-Yamabe problem on M is compact in C²(M) provided that convergence of the scalar curvature R[g+ ] of (X,g+ ) to -n(n+1) is sufficiently fast as ρ tends to 0 and the second fundamental form on M never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant to the geometric assumption imposed on X, our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
29. On a class of Kazdan-Warner equations.
- Author
-
Yu FANG and Mengjie ZHANG
- Subjects
RIEMANNIAN geometry ,SOBOLEV spaces ,CRITICAL point theory ,REAL numbers ,MATHEMATICAL models - Abstract
Let (Σ g) be a compact Riemannian surface without boundary and W1;2(Σ) be the usual Sobolev space. For any real number p > 1 and α ∊ R, we define a functional... has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu (DOI: 10.1007/s11425-017-9086-6). [ABSTRACT FROM AUTHOR]
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- 2018
- Full Text
- View/download PDF
30. A modified singular Trudinger–Moser inequality
- Author
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Wang, Yamin
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- 2019
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31. Regularity of area minimizing currents mod p
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Salvatore Stuvard, Jonas Hirsch, Andrea Marchese, Camillo De Lellis, University of Zurich, and Marchese, Andrea
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Regularity theory ,Current (mathematics) ,Area minimizing currents mod(p) ,Dimension (graph theory) ,340 Law ,610 Medicine & health ,Minimal surfaces, Area minimizing currents mod(p), Regularity theory, Multiple valued functions, Blow-up analysis, Center manifold ,Measure (mathematics) ,Set (abstract data type) ,Mathematics - Analysis of PDEs ,510 Mathematics ,Multiple valued functions ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,Center manifold ,Mathematics ,Discrete mathematics ,2603 Analysis ,Minimal surfaces ,Codimension ,10123 Institute of Mathematics ,Hausdorff dimension ,2608 Geometry and Topology ,Geometry and Topology ,Blow-up analysis ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure., 96 pages. Second part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Geom. Funct. Anal
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- 2020
32. Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures
- Author
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Aleks Jevnikar, Rafael López-Soriano, María Medina, and David Ruiz
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blow-up analysis ,Numerical Analysis ,Applied Mathematics ,Mathematics::Analysis of PDEs ,conformal metric ,Pohozaev-type identity ,Mathematics - Analysis of PDEs ,Prescribed curvature problem, conformal metric, blow-up analysis, Pohozaev-type identity ,FOS: Mathematics ,Mathematics::Differential Geometry ,Prescribed curvature problem ,35J20, 58J32, 53A30, 35B44 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
This paper is concerned with the compactness of metrics of the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and this new aspect is driven by the boundary condition., 31 pages, 1 figure
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- 2020
33. Semigroup and Blow-Up Dynamics of Attraction Keller-Segel Equations in Scale of Banach Spaces
- Author
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Iiyambo, David S. I. and Willie, Robert
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asymptotic dynamics ,blow-up analysis ,Keller-Segel model ,35B40-4 ,35K57 ,chemotaxis ,35K50 - Abstract
In this paper, we study the asymptotic and blow-up dynamics of the attraction Keller-Segel chemotaxis system of equations in scale of Banach spaces $E^\alpha_q = H^{2\alpha,q}(\Omega), −1 \le \alpha \le 1,1 \lt q \lt \infty$, where $\Omega \subset \mathbb{R}^N$ is a bounded spatial domain. We show that the system of equations is well-posed for a perturbed analytic semigroup, whenever $2\chi + a \lt \left( \frac{Ne\pi}{2} \right)^{\beta+\frac{\gamma}{2}-\frac{1}{2}}$, where $\chi$ is the chemical attractivity coefficient, $a$ is the rate of production of chemical, and $q, \beta, \gamma$ are of the scale spaces. Thus, as $t\nearrow\infty$, the asymptotic dynamics are captured in the limit set $\mathcal{M}\cup \{0\}$, where $\mathcal{M} = |\Omega|L^1 -$spatial average solutions. The constants for the sharp space embedding $E^\alpha_q \subset L ^\Theta(\Omega) (1\lt\Theta\le\infty)$ indicate that for either the application of Banach fixed point theorem, or the global existence of solutions, no need of either the time for a contraction mapping, nor the initial data of the system of equations, to be small, respectively. In blow-up dynamics, we prove that the solutions blow-up at the borderline scale spaces $E^\alpha_q, \alpha = \frac{N}{2q}$, independent of time $t > 0$, if the chemo-attractivity coefficient dominates the Moser-Trudinger threshold value. An analysis of the finite time bounds for blow-up of solutions in norm of $L^{2p}(\Omega),1 \le p \le 6$ and $\Omega \subset \mathbb{R}^N,N = 2, 3$, is also furnished.
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- 2018
34. Blow up analysis for a cosmic strings equation
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Gabriella Tarantello
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Mathematics::Analysis of PDEs ,Riemann sphere ,Context (language use) ,Conical singularities ,01 natural sciences ,symbols.namesake ,Singularity ,Mathematics - Analysis of PDEs ,Settore MAT/05 - Analisi Matematica ,FOS: Mathematics ,0101 mathematics ,Einstein ,Mathematics ,Liouville-type equations ,Plane (geometry) ,010102 general mathematics ,Mathematical analysis ,Blow-up analysis ,010101 applied mathematics ,Cosmic string ,Riemann hypothesis ,symbols ,Gravitational singularity ,35J15, 35J61 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new and surprising phenomena. For example in some cases, the blow up profile of the solution is described in terms of a differential problem that bares a geometrical meaning in the context of the uniformization of the Riemann sphere with conical singularities., 82 pages
- Published
- 2015
35. A Sharp Form of the Moser-Trudinger Inequality on a Compact Riemannian Surface
- Author
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Yang, Yunyan
- Published
- 2007
- Full Text
- View/download PDF
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