Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials

Abstract

Let $n\ge 2$, $0<\beta <n$, and let $V$ be a continuous nonnegative potential on ${\mathbb{R}}^n$ with a positive lower bound outside a ball. Under the constraint $\int_{\R^n}(|\nabla u|^n+V|u|^n)\dd\leq1$, we prove the singular Moser--Trudinger inequality with sharp coefficient $(1-\beta /n){\alpha }_n$, where ${\alpha }_n=n|{\mathbb{S}}^{n-1}{|}^{1/(n-1)}$. The subcritical supremum is attained for every such potential. If $V$ is radial and $r^{\beta }V(r)$ is nondecreasing, the critical supremum is also attained. This includes potentials that vanish on a ball and decrease to a positive limit at infinity. Under the same radial hypothesis, a bound for the tangential gradient shows that every extremal is radial and that the tangential gradients of maximizing sequences converge to zero in $L^n$, including at the critical coefficient. For the singular $n$-Laplacian equation with critical exponential growth, we obtain an explicit sufficient level for Palais--Smale compactness. Under the Ambrosetti--Rabinowitz condition and a smallness assumption at zero, this gives a nonzero nonnegative mountain-pass solution for all sufficiently large coupling parameters.