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

Authors: XIAOMENG LI

Abstract: 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 $0\leq\alpha<\lambda_{p}$. Using the method of blow-up analysis, we prove that for $p>2$ and $0\leq\alpha<\lambda_{p}$, the supremum \begin{align*}\sup_{u\in W^{1,2}(\mathbb{R}^2),\,\|u\|_{\alpha,p}\leq1}\int_{\mathbb{R}^2}(e^{4\pi u^2}-1-4\pi u^2)dx\end{align*} can be attained by some function $u_0\in W^{1,2}(\mathbb{R}^2)$ with $\|u_0\|_{\alpha,p}=1$.

Keywords: Trudinger--Moser inequality, extremal function, blow-up analysis

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