On Singular Trudinger–Moser Type Inequalities for Unbounded Domains and Their Best Exponents

Generalizations of the Trudinger–Moser inequality to Sobolev spaces with singular weights are considered for any smooth domain Ω???? ^N . Furthermore, we show that the resulting inequalities are sharp obtaining the best exponents.     

Sharp Weighted Young's Inequalities and Moser–Trudinger Inequalities on Heisenberg Type Groups and Grushin Spaces

We obtain sharp weighted Moser–Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for -symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form , for first-layer radial weights on a general Carnot group and functions with first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.Research supported by NSF grant DMS-0228807.     

Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

Potential Analysis - In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality...     

Trudinger–Moser Type Inequalities with Vanishing Weights in the Unit Ball

We introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion.

     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

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 R~n, wher~e n ≥ 2. More precisely, we show that for any given α 0 and 0 t n, then the following two inequalities hold for ■,■.We also consider the problem of the sharpness of the constant α_(n,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.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities

Let Ω be a smooth bounded domain in R~2, W_0~(1,2)(Ω) be the standard Sobolev space. By the method of energy estimate developed by Malchiodi–Martinazzi(J. Eur. Math. Soc., 16, 893–908(2014)), Mancini–Martinazzi(Calc. Var. Partial Differential Equations, 56, 94(2017)) and Mancini–Thizy(J. Differential Equations, 266, 1051–1072(2019)), we reprove the results of Carleson–Chang(Bull. Sci. Math., 110, 113–127(1986)), Flucher(Comment. Math. Helv., 67, 471–497(1992)), Li(Acta Math. Sin. Engl. Ser., 22, 545–550(2006)) and Su(J. Math. Inequal., in press). Namely, for any real number α≤ 1, the supremum ■ can be achieved by some function v ∈ W_0~(1,2)(Ω) with ║?_v║_2~2≤ 4π.     

The Neumann Problem on Unbounded Domains of ℝ d and Stochastic Variational Inequalities

ABSTRACT

An elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied in a space L ²(? ^d , ν) where ν is an invariant measure. The corresponding semigroup generated by the elliptic operator is identified with the transition semigroup associated with a stochastic variational inequality.     

Weighted Elliptic Systems of Lane–Emden Type in Unbounded Domains

We study the existence of weak solutions for a nonlinear elliptic system of Lane-Emden type $$\left\{\begin{array}{ll} -\Delta u \; = \; {\rm sgn}(v)|v|^{p-1} & {\rm in}\;\mathbb{R}^N, \\ -\Delta v \; = \; -\rho(x){\rm sgn}(u)|u|^{\frac{1}{p-1}} + f(x, u) & {\rm in}\;\mathbb{R}^N, \\ u, v \to 0 \quad {\rm as} \quad |x| \to +\infty, \end{array}\right.$$ by means of the Mountain Pass Theorem and some compact imbeddings in weighted Sobolev spaces.     

Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of $W^{2,n/2}(B_n)$ into the exponential class, where $B_n$ is the unit ball of ${\mathbb{R}}^n$. The corresponding sharp results for the spaces $W_0^{d,n/d}(\Omega )$ are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for $d=1$ and are due to Chang–Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the “canonical” solution of the Poisson equation on the ball, that is, the unique solution of the equation $\mathrm{\Delta }u=f$ which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function.     

On the Relationship between Sharp Kolmogorov-Type Inequalities and Sharp Kolmogorov–Remez-Type Inequalities

Ukrainian Mathematical Journal - We establish a new theorem on the relationship between sharp constants in Kolmogorov-type inequalities and sharp constants in Kolmogorov–Remez-type...     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

New Improved Moser�CTrudinger Inequalities and Singular Liouville Equations on Compact Surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern–Simons vortices in a self-dual regime. Using new improved versions of the Moser–Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.     

Attractors of the Derivative Complex Ginzburg-Landau Equation in Unbounded Domains

We consider the following initial boundary problem of derivative complex Ginzburg-Landau （DCGL） equation