Improved Moser-Trudinger inequality of Tintarev type in dimension \(n\) and the existence of its extremal functions
For a smooth bounded domain \(\Omega\subset \mathbb{R}^n\) with \(n\geq 2\), let \(W_0^{1,n} (\Omega)\) be the usual Sobolev space on \(\Omega\) and define \(\lambda_1 (\Omega) = \inf_{u\in W_0^{1,n} (\Omega)\setminus \{0\}} \frac{\int_{\Omega} |\nabla u |^n dx }{\int_{\Omega} |u|^n dx}\). Based on the blow-up analysis method, the author proves the following improved Moser-Trudinger inequality of Tintarev type \[ \sup_{u\in W_0^{1,n} , \int_{\Omega} |\nabla u |^n dx - \alpha \int_{\Omega} |u|^n dx \leq 1} \int_{\Omega} \exp (\alpha_n |u|^{\frac{n}{n-1}} ) dx < \infty\tag{1} \] for any \(0 \leq \alpha < \lambda_1 (\Omega)\), where \(\alpha_n = n \omega_{n-1}^{\frac{1}{n-1}}\) for the surface area \(\omega_{n-1}\) of the unit sphere in \(\mathbb{R}^n\). The inequality (1) is stronger than the improved Moser-Trudinger inequality obtained by \textit{Adimurthi} and \textit{O. Druet} [Commun. Partial Differ. Equations 29, No. 1--2, 295--322 (2004; Zbl 1076.46022)] in dimension 2 and by \textit{Y. Yang} [J. Funct. Anal. 239, No. 1, 100--126 (2006; Zbl 1141.46015)] in higher dimension and extends a result of \textit{C. Tintarev} [J. Funct. Anal. 266, No. 1, 55--66 (2014; Zbl 1303.26022)] in dimension 2 to higher dimension. Furthermore, the author proves that the supremun in (1) is attained for any \(0< \alpha < \lambda_1 (\Omega)\). The paper is systematically well-organized.
scientific article; zbMATH DE number 6970066
scientific article