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Register a new userOn a weighted Trudinger-Moser inequality in $mathbb{R}^N$
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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 $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, 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 Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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Let $W^{1,n} ( mathbb{R}^{n} $ be the standard Sobolev space and $leftVert cdotrightVert _{n}$ be the $L^{n}$ norm on $mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm [ underset{leftVert urightVert _{W^{1,n}left(mathbb{R} ^{n}right) }=1}{sup}int_{ mathbb{R}^{n}}Phileft( alpha_{n}leftvert urightvert ^{frac{n}{n-1}}left( 1+alphaleftVert urightVert _{n}^{n}right) ^{frac{1}{n-1}}right) dx<+infty ]in the entire space $mathbb{R}^n$ for any $0leqalpha<1$, where $Phileft( tright) =e^{t}-underset{j=0}{overset{n-2}{sum}}% frac{t^{j}}{j!}$, $alpha_{n}=nomega_{n-1}^{frac{1}{n-1}}$ and $omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $mathbb{R}^n$. We also show that the above supremum is infinity for all $alphageq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $alpha>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Phi(t)$ is replaced by a strictly smaller $Phi^*(t)=e^{t}-underset{j=0}{overset{n-1}{sum}}% frac{t^{j}}{j!}$. (Note that $Phi(t)=Phi^*(t)+frac{t^{n-1}}{(n-1)!}$).
We give a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces. In particular we can lower the smoothness requirement of the metric and apply the same approach to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.
Wang and Ye conjectured in [22]: Let $Omega$ be a regular, bounded and convex domain in $mathbb{R}^{2}$. There exists a finite constant $C({Omega})>0$ such that [ int_{Omega}e^{frac{4pi u^{2}}{H_{d}(u)}}dxdyle C(Omega),;;forall uin C^{infty}_{0}(Omega), ] where $H_{d}=int_{Omega}| abla u|^{2}dxdy-frac{1}{4}int_{Omega}frac{u^{2}}{d(z,partialOmega)^{2}}dxdy$ and $d(z,partialOmega)=minlimits_{z_{1}inpartialOmega}|z-z_{1}|$.} The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $mathbb{R}^{2}$ via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space $mathbb{B}={z=x+iy:|z|=sqrt{x^{2}+y^{2}}<1}$: [ sup_{|u|_{mathcal{H}}leq 1} int_{mathbb{B}}(e^{4pi u^{2}}-1-4pi u^{2})dV=sup_{|u|_{mathcal{H}}leq 1}int_{mathbb{B}}frac{(e^{4pi u^{2}}-1-4pi u^{2})}{(1-|z|^{2})^{2}}dxdy< infty, ] by using the method employed earlier by Lam and the first author [9, 10], where $mathcal{H}$ denotes the closure of $C^{infty}_{0}(mathbb{B})$ with respect to the norm $$|u|_{mathcal{H}}=int_{mathbb{B}}| abla u|^{2}dxdy-int_{mathbb{B}}frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].
In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $mathbb{B}$ be the unit ball in $mathbb{R}^N$ $(Ngeq 2)$, $p>1$, $g=|x|^{frac{2p}{N}beta}(dx_1^2+cdots+dx_N^2)$ be a conical metric on $mathbb{B}$, and $lambda_p(mathbb{B})=infleft{int_mathbb{B}| abla u|^Ndx: uin W_0^{1,N}(mathbb{B}),,int_mathbb{B}|u|^pdx=1right}$. We prove that for any $betageq 0$ and $alpha<(1+frac{p}{N}beta)^{N-1+frac{N}{p}}lambda_p(mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $uin W_0^{1,N}(mathbb{B})$ with $int_mathbb{B}| abla u|^Ndx-alpha(int_mathbb{B}|u|^p|x|^{pbeta}dx)^{N/p}leq 1$, there holds $$int_mathbb{B}e^{alpha_N(1+frac{p}{N}beta)|u|^{frac{N}{N-1}}}|x|^{pbeta}dxleq C,$$ where $|x|^{pbeta}dx=dv_g$, $alpha_N=Nomega_{N-1}^{1/(N-1)}$, $omega_{N-1}$ is the area of the unit sphere in $mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<beta<0$ and $alpha=0$ was considered by Adimurthi-Sandeep cite{A-S}, while the case $p=N=2$, $betageq 0$ and $alpha=0$ was studied by de Figueiredo-do O-dos Santos cite{F-do-dos}.
Suppose $F: mathbb{R}^{N} rightarrow [0, +infty)$ be a convex function of class $C^{2}(mathbb{R}^{N} backslash {0})$ which is even and positively homogeneous of degree 1. We denote $gamma_1=inflimits_{uin W^{1, N}_{0}(Omega)backslash {0}}frac{int_{Omega}F^{N}( abla u)dx}{| u|_p^N},$ and define the norm $|u|_{N,F,gamma, p}=bigg(int_{Omega}F^{N}( abla u)dx-gamma| u|_p^Nbigg)^{frac{1}{N}}.$ Let $Omegasubset mathbb{R}^{N}(Ngeq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0leq gamma <gamma_1$, we have $$ sup_{uin W^{1, N}_{0}(Omega), |u|_{N,F,gamma, p}leq 1}int_{Omega}e^{lambda |u|^{frac{N}{N-1}}}dx<+infty, $$ where $0<lambda leq lambda_{N}=N^{frac{N}{N-1}} kappa_{N}^{frac{1}{N-1}}$ and $kappa_{N}$ is the volume of a unit Wulff ball. Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any $0 leqgamma <gamma_1$.
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