Remarks on the Extremal Functions for the Moser–Trudinger Inequality

Abstract

We will show in this paper that if λ is very close to 1, then

$$ I(M,\lambda ,m) = {\mathop {\sup }\limits_{u \in H^{{1,n}}_{0} (M),\smallint _{M} {\left| {\nabla u} \right|}^{n} dV = 1} }{\int_\Omega {{\left( {e^{{\alpha _{n} {\left| u \right|}^{{\frac{n} {{n - 1}}}} }} - \lambda {\sum\limits_{k = 1}^m {\frac{{{\left| {\alpha _{n} u^{{\frac{n} {{n - 1}}}} } \right|}}} {{k!}}} }} \right)}dV} } $$

can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" (Comm. Pure. Appl. Math., 55, 135–152, 2002).