Shiqiu Fu, Kanishka Perera
We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.
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The main aim of this article is to study non-singular version of Moser-Trudinger and Adams-Moser-Trudinger inequalities and the singular version of Moser-Trudinger equality in the Cartesian product of Sobolev spaces. As an application of these inequalities, we study a system of Kirchhoff equations with exponential non-linearity of Choquard type.
Sharp form for improved Moser-Trudinger inequality
证明了具有最优系数$α=1/2$的改进的Moser-Trudinger不等式对$S^2$上所有具有零矩的函数都成立.
We prove the existence of positive solutions for a class of semipositone problem with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.
Extremal function for a sharp Moser-Trudinger type inequality on the upper half space
本文在二维上半空间建立新的尖锐Moser-Trudinger型不等式，并证明单位球内动态变化下该类型不等式极值函数的存在性，为非线性PDE与几何研究提供工具。
In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.
Extremal Functions for the Singular Moser-Trudinger Inequality in 2 Dimensions
在这篇文章中，我们证明了一个奇异的Moser-Trudinger不等式，由于Adimurthi-Sandeep，在2维极值函数的存在性。
Remarks on the Moser-Trudinger inequality
本文将Moser-Trudinger不等式推广到任何满足Poincaré不等式的欧几里得整环。我们发现，同样的等价性一般不适用于单位球上的共形度量，给出了反例。我们还研究了无界域上Moser-Trudinger不等式极值的存在性，证明了无限平面带上的极值存在性。
On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem
本文在整个欧几里得空间上建立了一个具有完全Sobolev范数约束的加权Trudinger-Moser型不等式.主要工具是最近由Adimurthi和Yang建立的关于整个空间的奇异Trudinger-Moser不等式，以及函数的变换。我们还讨论了相关的变分问题的极大化的存在性和不存在性。
Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$
本文研究整条实直线上分数索伯列夫空间$H^{1/2,2}$中函数的临界和次临界Trudinger-Moser型不等式.证明了两个不等式之间的关系，并讨论了上确界的可达性。
A simplification of the proof of the existence of the extremal function for the Moser-Trudinger inequality
在前一篇文章中，作者证明了紧流形上Moser-Trudinger不等式极值函数的存在性。本文将给出其中一个关键命题的新证明。
In this paper, using the method of blow-up analysis, we obtained a Trudinger-Moser inequality involving L p -norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger-Moser functional. This extends an early result of Yang [27]. Similar result in the Euclidean space was also established by Lu-Yang [17].
We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,α}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.
In this paper, we concern trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. This kind of inequalities were extensively studied by Osgood-Phillips-Sarnak [24], Liu [20], Li-Liu [17], Yang [31, 32] and others. We establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals via the method of blow-up analysis. The results in the current paper generalize those of Li-Liu [17] and Yang [32, 33].
Moser functions and fractional Moser-Trudinger type inequalities
本文改进了一些分数阶Moser-Trudinger型不等式的尖锐性，特别是Lam-Lu和Martinazzi的研究.作为应用，改进了Adimurthi和Lakkis的工作，证明了Dirichlet边值问题$(-Δ)^\frac{n}{2}u=λue^{bu^2} \,\text{ in }Ω,\, 0<λ<λ_1,\,b>0,$的弱解的存在性，对于有限测度的$\mathbb{R}^n$中的任意区域$(-Δ)^\frac{n}{2}u=λue^{bu^2} \,\text{ in }Ω,\, 0<λ<λ_1,\,b>0,$.这里$λ_1$是$Ω$上的$(-Δ)^\frac n2$的第一特征值。
We prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp estimates obtained via commutator techniques.
A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one
本文在Sobolev-Slobodeckij空间的背景下讨论了Parini和Ruf关于一类Moser-Trudinger型不等式的最新结果.我们进一步推进他们的分析考虑不平等的整体$\mathbb{R}$，我们给他们的一个开放的问题的答案。
Bifurcation results for problems with fractional Trudinger-Moser nonlinearity
通过使用一个合适的拓扑参数的基础上同调链接和利用Trudinger-Moser不等式在分数空间最近获得的，我们证明了多个解决方案的存在性问题涉及的非线性分数拉普拉斯算子和相关的临界指数非线性。这扩展了文献中的N-Laplacian算子的结果。
New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces
我们考虑一个奇异的刘维尔方程的紧凑的表面上，所产生的研究的Chern-Simons涡在一个自对偶制度。使用新的改进版本的Moser-Trudinger不等式（其主要特点是尺度不变）和变分方案，我们证明了新的存在性结果。
Moser-Trudinger inequalities for singular Liouville systems
本文证明了一般奇异Liouville系统的Euler-Lagrange泛函的Moser-Trudinger不等式。我们的特点的参数值，产生的功能，我们从下面给出有界性的必要条件。在对系统系数的某些假设下，我们还给出了一个尖锐的不等式。
Extremal Functions for Singular Moser-Trudinger Embeddings
研究了奇异势存在下的Moser-Trudinger型泛函。特别地，我们利用Onofri不等式对$\mathbb{R}^2$中的单位圆盘给出了一个奇异Carleson-Chang型估计的证明.此外，我们考虑了具有圆锥奇点的紧致曲面上的Adimurthi-Druet型泛函，并讨论了这类泛函极值的存在性，推广了Castleman和Roy以前的结果。
Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity
研究了一类定义在整条实数线上的非线性标量场方程基态解的存在性，其中包含分数Laplacian算子和Trudinger-Moser临界增长的非线性项.我们处理缺乏紧凑性的相关能量泛函由于域的无界性和存在的极限情况下嵌入。
A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space
本文给出了$\mathbb{R}^n$上次临界Trudinger-Moser不等式的一个新证明.现有的关于这个不等式的证明都是基于索伯列夫空间$W^{1,n}(\mathbb{R}^n)$中函数的重排论证。我们的方法避免了这种技术，因此可以用于黎曼流形的情况下，并在整个海森堡群。
Moser-Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension 2
设$(M,g)$是二维紧致黎曼流形.本文利用爆破分析的方法证明了$(M,g)$上向量丛的几个Moser-Trdinger型不等式.在爆破发生的假设下，我们还得到了这类不等式的一个上界。
Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions
本文在全空间分数维情形下建立临界与次临界尖锐Trudinger-Moser不等式，通过渐近上下界证明二者等价，并由此得到极值函数的存在性及特殊情形下临界上确界的显式值。
In this article, we firstly study the cone Moser-Trudinger inequalities and their best exponents $α_2$ on both bounded and unbounded domains $\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study the existence of weak solutions to the nonlinear equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{\mathbb{B}} u=f(x, u), &\mbox{in}\ x\in \mbox{int} (\mathbb{B}), \\ u= 0, &\mbox{on}\ \partial\mathbb{B}, \end{array} \right. \end{equation*} where $Δ_{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_1 =0…
A Moser-Trudinger inequality for the singular Toda system
在本文中，我们证明了一个尖锐的版本的Moser-Trudinger不等式的Euler-Lagrange功能的奇异Toda系统，在Chern-Simons理论模型的研究的启发。我们的结果扩展了标量的情况下，以及为经常户田系统。我们希望这个不等式是一个基本的工具，攻击变分一般假设下的存在性问题。
Trudinger-Moser inequalities on the entire Heisenberg group
继续我们以前的工作（Cohn，Lam，Lu，Yang，Nonlinear Analysis（2011），doi：10.1016 /j.na.2011.09.053），我们得到了整个海森堡群上的一类Trudinger-Moser不等式，它指示了最佳常数是什么。在欧几里得空间或海森堡群的无界区域上，已有的类似不等式的证明都是基于重排论证。在本文中，我们提出了一种新的方法来解决这个问题。具体地说，我们得到了整体Trudinger-Moser不等式的帮助下，截断函数的粘合局部估计。我们的方法仍然适用于类似的问题时，海森堡群被Eclidean空间或完全非紧黎曼流形。
This is a continuation of our previous work [13]. Let $(Σ,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on $(Σ,g)$. Trudinger-Moser inequalities involving $\mathbf{G}$ are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen [7], Iula-Manicini [21], and the authors [13].
We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Apéry's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.
In this paper, we derive a sharp version of the singular Trudinger-Moser inequality, which was originally established by Adimurthi and Sandeep (Nonlinear Differ. Equ. Appl. 2007). Moreover, extremal functions for those singular Trudinger-Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work (J. Differential Equations 2015), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behaviour of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classi…
On a supercritical k-Hessian inequality of Trudinger-Moser type and extremal functions
该研究针对ℝ^{N}中单位球上的k-容许径向对称函数空间，建立了k-Hessian算子的超临界Trudinger-Moser型不等式，并证明了对应极值函数的存在性，拓展了相关不等式理论。
The Moser-Trudinger inequality and its extremals on a disk via energy estimates
研究了单位圆盘上Moser-Trudinger不等式的非负径向对称临界点$u_μ$在$\mathbb{R}^2$中的Dirichlet能量，证明了它可展开为$$4π+\frac{4π}{μ^{4}}+o(μ^{-4})\le \int_{B_1}|\nabla u_μ|^2dx\le 4π+\frac{6π}{μ^{4}}+o(μ^{-4}),\quad \text{as }μ\to\infty,$$，其中$μ=u_μ(0)$是$u_μ$的最大值.因此，我们得到了Moser-Trudinger不等式的新证明，Carleson-Chang关于极值存在性的结果，以及Struwe和Lamm-Robert-Struwe多重性在超临界状态下的结果（仅在单位圆盘的情况下）。我们的结果是稳定的充分弱扰动下的Moser-Trudinger功能。我们明确确定的临界水平的扰动，虽然扰动Moser Trudinger不等式仍然成立，其临界点的能量收敛到$4π$从下面。我们预计，在某些情况下，极值的存在并不成立，也不存在超临界状态下的临界点。
Sharp Moser-Trudinger inequalities on Riemannian manifolds with Negative curvature
本文在负曲率完备单连通黎曼流形上建立了具有尖锐常数的Moser-Trudinger不等式。
New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D
本文综述二维Trudinger-Moser不等式，从几何问题出发探讨Onofri不等式与欧氏区域精确不等式的联系，并介绍含对数核的非局部相互作用能量泛函的最新结果。
On compactness in the Trudinger-Moser inequality
本文研究了二维Moser非线性方程在弱收敛条件下的连续性。与Sobolev不等式中的临界非线性不同，Sobolev不等式中的临界非线性在任何点都缺乏弱连续性，Moser泛函只有在Moser族中的一个集中函数序列的例外情况下才不是弱连续的（直到平移，其余的在Sobolev范数中消失）。该论点是基于弱收敛缺陷的结构描述，类似于Solimini为Sobolev不等式建立的剖面分解，但涉及特定于二维情况的规范算子。
Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) …
In this short note, we generalized an energy estimate due to Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to reprove existence of extremals for Trudinger-Moser inequalities of Adimurthi-Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-θ}(r^α\vert u'(r)\vert^βu'(r))'$, where $θ, β\geq 0$ and $α>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Pólya-Szeg{ö} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
A Hardy-Moser-Trudinger inequality
本文得到了平面上单位圆盘$B$上的一个不等式，同时改进了经典的Moser-Trudinger不等式和经典的Hardy不等式。也就是说，存在一个常数$C_0>0$，使得\[ \int_B e^{\frac {4πu^2}{H(u)}} dx \le C_0 < \infty, \quad \forall\; u\in C^\infty_0(B),\]，其中$$H(u) := \int_B |\n u|^2 dx - \int_B \frac {u^2}{(1-|x|^2)^2} dx.$$这个不等式是最近被深入研究的高维Hardy-Sobolev-Maz'ya不等式的一个二维类似.我们还证明了上确界是在一个合适的函数空间中得到的，这类似于Carleson-Chang关于Moser-Trudinger不等式的著名结果.
Improved Lerey inequality and Trudinger-Moser type inequality involving the Leray potential
该研究以径向对称函数分析为基础，改进了含Leray位势的Leray不等式，引入相关Hilbert空间并讨论嵌入性质，还得到了二维单位球中对应的Trudinger-Moser型不等式。
Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains
该研究针对无界区域上带位势权的Sobolev空间，建立其到径向幂权加权Lebesgue空间的嵌入关系，明确对应Trudinger-Moser型不等式极值的存在性与不存在性，并优化了嵌入的最大可积性。
Concentration profiles for the Trudinger-Moser functional are shaped like toy pyramids
本文回答了Adimurthi和Struwe关于半线性Trudinger-Moser泛函（以及具有更一般临界非线性度的泛函）在除n/2以外的所有水平上都满足Palais-Smale条件的猜想.在本文中，我们构建临界序列在任何水平大于大于1/2对应于一个大家庭的不同的浓度分布，索引的所有封闭的子集C（0，1），出现在二维的情况下，而不是“标准泡沫”在更高的维度。该方法是基于Solimini风格的剖面分解。
Is the Trudinger-Moser nonlinearity a true critical nonlinearity?
当Sobolev空间$H^1$在$N>2$维上的临界非线性度$\int |u|^{2^*}$在任何点上都缺乏弱连续性时，Trudinger-Moser非线性度$\int e^{4πu^2}$在$N=2$维上在除零以外的任何点上都是弱连续的。在前一种情况下，弱连续性的缺乏可以归因于平移和扩张作用的不变性。单位圆盘$\mathbb D\subset\R^2$的Sobolev空间$H_0^1$具有类似于平移的变换（莫比乌斯变换）和非线性膨胀$r\mapsto r^s$。我们提出了改进的Trudinger-Moser不等式，具有比$\int e^{4πu^2}$更尖锐的非线性，在任何点都缺乏弱连续性，并具有（分别），平移和伸缩不变性。然而，我们发现，没有非线性的形式$\int F(|x|,u(x))\mathrm{d}x$是既膨胀和莫比乌斯位移不变。本文还给出了Mancini和Sandeep最近得到的共形不变Trudinger-Moser不等式和Beckner的Onofri型不等式的一个更尖锐的新的、非常简短的证明。
Moser-Trudinger inequalities: from local to global
该研究引入完备黎曼流形上的局部Moser-Trudinger不等式概念，通过Poincaré不等式或更强范数条件将其从局部推广至全局，应用于Hadamard流形，且技术可推广至Heisenberg群等次黎曼情形。
A weighted Moser-Trudinger inequality and its relation to the Caffarelli-Kohn-Nirenberg inequalities in two space dimensions
首先证明了二维欧几里得空间中一个依赖于参数的Moser-Trudinger型加权不等式。如果参数大于-1，则不等式对径向函数成立。在没有对称性假设的情况下，当且仅当参数在区间（-1，0]内时，它成立。不等式给了我们一些洞察力的对称性破缺现象的极值功能的Hardy-Sobolev不等式，建立了由Caffarelli-Kohn-Nirenberg，在两个空间维度。事实上，对于合适的参数集（渐近尖锐），我们证明对称性或对称性破缺的爆破方法。这样，加权Moser-Trudinger不等式就成为Hardy-Sobolev不等式的一个极限情形。
Remarks on the Extremal Functions for the Moser-Trudinger Inequalities
本文证明了如果$λ$非常接近于1，那么可以得到$$I(M,λ,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_Ω(e^{α_n |u|^\frac{n}{n-1}}-λ\sum\limits_{k=1}^m\frac{|α_nu^\frac{n}{n-1}|^k} {k!})dV,$$，其中$M$是一个紧的有边带边流形.这个结果给出了一个反例的猜想de Figueiredo，do ó，和Ruf在他们的论文题为“在一个不等式由N.Trudinger和J.Moser和相关的椭圆型方程“（通讯纯。应用数学，{ \bf 55}：135-152，2002）。
Trudinger-Moser inequality with remainder terms
本文给出了Trudinger-Moser不等式的一个改进，其中的约束集不是由平方梯度范数定义的，而是由平方梯度范数减去一个加权Lp型余项定义的.这是Hardy-Sobolev-Mazya不等式在高维中的二维对应，它是极限Sobolev不等式的类似改进。特别地，我们推广了Adimurthi和Druet以及Wang和Ye发现的两种已知的势型余项（即加权L^2-项）。此外，我们还证明了带有L^p-余项（p>2）的不等式，并对Onofri-Beckner不等式给出了类似的改进.
Optimal Trudinger-Moser inequalities on complete noncompact Riemannian manifolds: Revisit of the argument from the local inequalities to global ones
本文通过严谨化和提供替代方法，探讨了从局部到全局的Trudinger-Moser不等式论证过程。
Best Constants for Anisotropic Moser-Trudinger Inequalities with the Exact Growth Condition and Their Applications
该研究在$\boldsymbol{\text{R}}^n$的Finsler度量框架下，建立了精确增长条件下各向异性Moser-Trudinger不等式的最佳常数，引入拟共形型变换推导新变体，应用其分析$n$-Finsler-Laplace方程正基态解的存在性。
We propose a power type approximation of the Moser--Trudinger functional and show that its concentration level converges to the Carleson--Chang limit.
A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N
本文研究N维对数核Trudinger-Moser型不等式，发现依赖N的尖锐阈值决定极值函数存在或爆破，并证明极值函数满足Euler-Lagrange方程，将结果推广到任意N>2。
Moser-Trudinger inequality on conformal discs
我们证明了在共形圆盘中Moser-Trudinger不等式成立当且仅当该度量由双曲度量从上有界。在二维欧几里得空间的无界子集上，我们还得到了Moser-Trudinger不等式成立的一个充要条件.
Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications
本文在带势权重的Sobolev空间上建立了尖锐Sobolev嵌入和带尖锐常数的Adams型不等式，并利用这些结果证明了具有多项式与指数增长非线性的椭圆方程存在非平凡解。
Extremal function for Moser-Trudinger type Inequality with Logarithmic weight
在加权径向索伯列夫空间上，Calanchi和Ruf在2维空间上建立了Moser-Trudinger型不等式的如下推广：如果$β\in [0,1)$和$w_0(x) = |\log |x||^β$则$$ \sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{αu^{\frac{2}{1-β}}} dx < \infty,$$当且仅当$α\leq α_β= 2\left[2π(1-β) \right]^{\frac{1}{1-β}}.$我们证明了当$α= α_β$时上述不等式在临界情形下极值函数的存在性，从而推广了Carleson-Chang证明当$β=0$时的结果。
$(N,q)$-Laplacian problems with critical Trudinger-Moser nonlinearities
我们得到了一个有界区域中具有临界Trudinger-Moser非线性的$(N,q)$ -Laplacian问题的非平凡解。除了与临界非线性问题相关的紧致性损失的通常困难之外，这个问题缺乏适合于应用经典连接定理的直和分解。我们表明，每一个Palais-Smale序列在低于一定的能量阈值的水平承认一个子序列，弱收敛到一个非平凡的变分泛函的临界点。然后，我们证明了一个抽象的临界点定理的基础上的上同调指标，并使用它来构建一个minimax水平低于这个阈值。
Druet [6] proved that if $(f_γ)_γ$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_γ)_γ$ solves $$ \begin{cases} &Δu =f_γ(x,u)\,,~~ u>0\text{ in }Ω\,,\\ &u =0\text{ on }\partialΩ\,, \end{cases} $$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_γ$ converges to $4πN$ plus the Dirichlet energy of $u_\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities…
We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -Δu=λue^{u^2}~~&\mbox{in}~Ω,\\[0.5mm] u>0 ~~ &{\text{in}~Ω},\\[0.5mm] u=0~~&\mbox{on}~\partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $λ>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of lo…
The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $Ω\subset\mathbb{R}^n$ is bounded, $ω_{n-1}$ is the $\mathcal{H}^{n-1}$ measure of the unit sphere, then for $α>0$ and $β\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_Ω\frac{e^{α|u|^{n/(n-1)}}}{|x|^β}\leq C \ \Leftrightarrow \ \fracα{α_n}+\fracβ{n}\leq1,\qquad $$ where $α_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(Ω) \ | \ \i…
Combining Carleson-Chang's result 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, extended by the author to high dimensional case and Riemannian surface case, generalized by C. Tintarev to wider cases including singular form and by M. de Souza and J. M. do Ó 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…
In this paper we prove the pluricomplex counterpart of the Moser-Trudinger and Sobolev inequalities in complex space. We consider these inequalities for plurisubharmonic functions with finite pluricomplex energy, and we estimate the concerned constants.
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