An Improved Leray-Trudinger Inequality

Abstract

In this article, we have derived the following Leray-Trudinger type inequality on a bounded domain $\Omega$ in ${\mathbb{R}}^n$ containing the origin. \begin{align*} \displaystyle{\sup_{u\in W^{1,n}_{0}(Ω), I_{n}[u,Ω,R]\leq 1}}\int_Ω e^{c_n\left(\frac{|u(x)|}{E_{2}^β(\frac{|x|}{R})}\right)^{\frac{n}{n-1}}} dx < +\infty \ \text{, for some } c_n>0 \ \text{depending only on } n. \end{align*} Here $\beta =\frac{2}{n}$, $I_n[u,\Omega ,R]:={\int }_{\Omega }|\nabla u{|}^{n}dx-{\left(\frac{n-1}{n}\right)}^n{\int }_{\Omega }\frac{|u{|}^n}{|x{|}^n{E}_1^n(\frac{|x|}{R})}dx$, $R\ge \underset{x\in \Omega }{\sup }|x|$ and $E_1(t):=\log (\frac{e}{t})$, $E_2(t):=\log (e{E}_1(t))$ for $t\in (0,1].$ This improves an earlier result by Psaradakis and Spector. Also we have proved that, for any $c>0$ the above inequality is false, if we take $\beta <\frac{1}{n}.$

11 pages