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

Özet

Öz Let $W^{1,2} \mathbb{R}^2 $ be the standard Sobolev space. Denote for any real number $p>2$ \begin{align*}\lambda_{p}=\inf\limits_{u\in W^{1,2} \mathbb{R}^2 ,u\not\equiv0}\frac{\int_{\mathbb{R}^{2}} |\nabla u|^2+|u|^2 dx}{ \int_{\mathbb{R}^{2}}|u|^pdx ^{2/p}}. \end{align*} Define a norm in $W^{1,2} \mathbb{R}^2 $ by \begin{align*}\|u\|_{\alpha,p}=\left \int_{\mathbb{R}^{2}} |\nabla u|^2+|u|^2 dx-\alpha \int_{\mathbb{R}^{2}}|u|^pdx ^{2/p}\right ^{1/2}\end{align*} where $0\leq\alpha2$ and $0\leq\alpha