A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in \(\mathbb R^2\)
From the abstract: Let \(\Omega \subset \mathbb{R}^2\) be a smooth bounded domain, and \(q(x)\) be a polynomial with \(q(0) \neq 0\). Then under some hypothesis on \(q(x)\), there holds \[ \sup_{\int_{\Omega} |\nabla u|^2\mathrm{d}x=1,\,\int_{\Omega}u\mathrm{d}x =0} \int_{\Omega} \mathrm{e}^{\frac{2\pi u^2}{q(0)}q(\int_{\Omega}u^2\mathrm{d}x) }\mathrm{d}x < \infty. \] A sufficient condition will be given to assure that the above inequality does not hold. Furthermore, the existence of the extremal functions will be derived.
scientific article; zbMATH DE number 5553053