Heisenberg群中带奇异权的最优临界Hardy-Trudinger-Moser不等式

本文建立了Heisenberg群中有界域和一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.克服临界Hardy不等式和奇异权函数带来的困难,利用带奇异权的Trudinger-Moser不等式和一些基本估计建立了有界域上一般的带奇异权的临界Hardy-Trudinger-Moser不等式,并通过选取适当的Moser函数得到了最佳常数.最后,利用分割积分区域的方法得到了一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.

负幂次对数加权的Trudinger-Moser不等式

本文研究具有负幂次对数加权的Trudinger-Moser不等式.通过建立了一个径向引理,利用著名的Leckband泛函不等式证明了Calanchi和Ruf(2015)得到的对数加权Trudinger-Moser不等式在负幂次情形依然成立.

Cartan-Hadamard流形上关于Lorentz范数的Trudinger-Moser不等式

本文研究了Cartan-Hadamard流形上带Lorentz范数的Trudinger-Moser不等式.利用了相关格林函数的逐点估计以及O’Neil不等式,我们得到了该不等式的最佳常数,推广了相应欧氏空间上的结果.

一维直线上的奇异型Trudinger-Moser不等式

本文研究了一维直线上的奇异型Trudinger-Moser不等式.利用分数次Sobolev空间上函数的Green表示公式,得到了一类奇异型Trudinger-Moser不等式.进一步利用合适的测试函数序列验证了不等式中常数的最佳性.这一结果将高维空间上的奇异型Trudinger-Moser不等式推广到了一维情形.

二维空间上对数加权各向异性范数约束下的Trudinger-Moser不等式

本文研究二维空间上一类各向异性对数加权径向Sobolev空间上的Trudinger-Moser不等式.通过建立一个重要的径向引理,并利用著名的Leckband泛函不等式得到了对数加权约束下的最佳Trudinger-Moser增长指标,特别地,我们得到在极限情形β=1下,Trudinger-Moser最佳增长为双指数形式增长.通过构造合适的测试函数序列证明了对数加权的Trudinger-Moser不等式中常数的最佳性.

Trudinger-Moser不等式的两种证明方法

Trudinger-Moser不等式在研究带有临界指数增长非线性项偏微分方程解的存在性问题上有着重要的应用.在径向空间中利用施瓦兹对称重排,基于单位分解的技巧取截断函数是证明Trudinger-Moser不等式的两种主要方法.

Extremal Functions for an Improved Trudinger-Moser Inequality Involving L^(P)-Norm in R^(n)

Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).

A Remark on Hardy-Trudinger-Moser Inequality

Let IB be the unit disc in R^2,H be the completion of C∞0(B)under the norm||u||H=(∫B|▽u|^2dx-∫Bu^2/(1-|x^2|^2dx)^1/2,■u∈C∞0(B).Using blow-up analysis,we prove that for anyγ≤4π,the supremum sup u∈H,||u||1,h≤1∫Beγu^2dx can be attained by some function u0∈H with||u0||1,h=1,where is a decreasingly nonnegative,radially symmetric function,and satisfies a coercive cond让ion.Namely there exists a constantδ>0 satisfying||u||21,h=||u||2H-∫Bhu^2dx≥δ||u||H^2,■u∈H.This extends earlier results of Wang-Ye[1]and Yang-Zhu[2].

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.

A New Proof of Subcritical Trudinger-Moser Inequalities on the Whole Euclidean Space

In this note, we give a new proof of subcritical Trudinger-Moser inequality on R^n. All the existing proofs on this inequality are based on the rearrangement ar-gument with respect to functions in the Sobolev space W^1,n （R^n）. Our method avoids this technique and thus can be used in the Riemamlian manifold case and in the entire Heisenberg group.

Finsler Trudinger-Moser inequalities on R^(2)

The first aim of this article is to study the sharp singular(two-weight)Trudinger-Moser inequalities with Finsler norms on R^(2).The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions.Finally,by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms,we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient.

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W^(1,2)(Σ,g),∫_(θΣ)uds_(g)=0,u≠0∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)/∫_(θΣ)u^(2)ds_(g)and H={u∈W^(1,2)(Σ,g):∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)-α∫_(θΣ)u^(2)ds_(g)≤1 and∫_(θΣ)uds_(g)=0},where W^(1,2)(Σ,g)denotes the usual Sobolev space and▽g stands for the gradient operator.By the method of blow-up analysis,we obtain sup_(u∈H)∫_(θΣ)e^(πu^(2))ds_(g){<+∞,0≤α﹤λ_(1)(∂Σ),=+∞,α≥λ_(1)(∂Σ)Moreover,the author proves the above supremum is attained by a function u∈H∩C^(∞)(∑)for any 0≤α<λ_(1)(θΣ).Further,he extends the result to the case of higher order eigenvalues.The results generalize those of[Li,Y.and Liu,P.,Moser-Trudinger inequality on the boundary of compact Riemannian surface,Math.Z.,250,2005,363–386],[Yang,Y.,Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J.Math.,227,2006,177–200]and[Yang,Y.,Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two,J.Diff.Eq.,258,2015,3161–3193].

全空间上与Trudinger-Moser-Lorentz不等式相关的集中紧性原理

本文研究了全空间上与Trudinger-Moser-Lorentz不等式相关的集中紧性原理.利用函数的水平截断方法,我们将有界区域上与Trudinger-Moser-Lorentz不等式相关的集中紧性原理推广到了无界区域上.此外,我们还构造了试验函数验证了结论的最佳性.

Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n.

A Weak Trudinger-Moser Inequality with a Singular Weight on a Compact Riemannian Surface

Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).

Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint

In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-1,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx≤c0|I|),andβq is optimal in the sense that u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx+∞),for anyβ>βq.Furthermore,when q is even,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ih(u)eβq|u(x)|q')dx≤+∞),for any function h:[0,∞→[0,∞)with lim t→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.

A Weighted Singular Trudinger-Moser Inequality

In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).

Extremal Functions for Trudinger-Moser Type Inequalities in RN

Let N≥2, aN=Nω1 N-1（N-1） , where ωN-1 denotes the area of the unit sphere in RN. In this note, we prove that for any 0 〈a〈aN and any β 〉 0, the supremum sup u∈W 1,N（R N）,｜｜u｜｜W 1,N（R N）≤1 ∫ R N ｜u｜β （e a ｜u｜ N/N-1 N-2 ∑j=0 aj/j!｜u｜ Ni/N-1）dx.can be attained by some function u ∈ W1, N （R N） with ｜｜ u ｜｜ W 1,N （R N ） = 1. Moreover, when a ≥ aN, the above supremum is infinity.

A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space H^n：u∈W^1,n（H^n）,^sup,fH^n｜ H^nu｜^ndμ≤1∫H^n ea｜u｜n/n-1-∑^n-2a^k｜u｜nk/n-1 k=0 k!/ρβ dμ〈∞ a/an＋β/n≤1,where α 〉 0,β E [0,n）, ρ and dμ are the distance function and volume element of H^n respectively.