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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.
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
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 approac
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}(Om
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 $mathbb{B}$
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_{Om