Moser--Trudinger inequality for functions with mean value zero (Q876950)

The present paper is devoted to the Moser--Trudinger inequality for functions with mean value zero. The author shows that \[ \sup\Biggl\{\int_\Omega e^{\alpha|u|^{{d\over d-1}}}\,dx : u\in H^{1,d}(\Omega),\;\int_\Omega |\nabla u|^d \,dx= 1,\;\int_\Omega u\,dx= 0\Biggr\} \] is attained for any \(\alpha\leq \alpha_d= d({\omega_{d-1}\over 2})^{{1\over d-1}}\), and the supremum is infinite for any \(\alpha> \alpha_d\), where \(\omega_{d-1}\) is the area of the unit sphere in \(\mathbb{R}^d\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) \((d\geq 2)\).