An improved Trudinger–Moser inequality and its extremal functions involving $L^p$ -norm in $mathbb{R}^2$

Abstract

Let $;W^{1,;2}(mathbb{R}^2)$ be the standard Sobolev space. Denote for any real number p > 2 $lambda p=;begin{array}{c}infuin W^{1,2}(mathbb{R}^2),unotequiv0end{array}$ $frac{int_{mathbb{R}^2}(vertnabla uvert^2+vert uvert^2)dmathcal x}{left(int_{mathbb{R}^2}vert uvert;rho dmathcal xright)^{2/rho}}$Define a norm in $;W^{1,;2}(mathbb{R}^2)$ by${parallel uparallel}_{alpha,p};=;;(int_{mathbb{R}^2}(vertnablaupsilonvert^2+vert uvert^2)dmathcal x-alpha{(int_{mathbb{R}^2}uvert^rho dmathcal x)}^{2/p})^{1/2}$where 0 ≤ α 2 and 0 ≤ α < λp , the supremum$begin{array}{c}supuin W^{1,2;}(mathbb{R}^2),;{parallel uparallel}_{alpha,p}leq1int_{mathbb{R}^{2^{(e^{4pi u2};-;1;-;4pi u^2;)dmathcal x}}}end{array}$can be attained by some function $u_0in W^{1,2}(mathbb{R}^2)withparallel u_0;parallel_{alpha,p}=;1$