Extremal for a k-Hessian inequality of Trudinger–Moser type

Abstract

We consider k-Hessian operators $S_k[u]$ in bounded domains $\varOmega$ in $\mathbb R^N$ such that $\partial \varOmega$ is $(k-1)$-convex. For so-called k-admissible functions $u \in \varPhi _0^k$ one has Sobolev type inequalities of the form

$$\begin{aligned} \Vert u\Vert _{L^p(\varOmega )} \le C\, \Vert u\Vert _{\varPhi _0^k} \end{aligned}$$

where $\Vert u\Vert _{\varPhi _0^k}^{k+1} = \int _\varOmega (-u) S_k[u]dx$, and $1 \le p \le k^* = \frac{N(k+1)}{N-2k}$. The case $N = 2k$ is a borderline case of Trudinger–Moser type, and it was recently shown by Tian–Wang that a corresponding inequality of exponential type holds

$$\begin{aligned} \sup _{\Vert u\Vert _{\varPhi _0^k \le 1}} \int _\varOmega \left( \mathrm {e}^{\alpha |u|^{\frac{N+2}{N}}} -\sum _{j=0}^{k-1}\frac{\alpha ^{j} |u|^{j\frac{N+2}{N}}}{j!} \right) \mathrm {d}x \le C \end{aligned}$$

for $\alpha \le \alpha _N = N\left[ \frac{\omega _{N-1}}{k} {N-1\atopwithdelims ()k-1}\right] ^{2/N}$. In this article we prove an analogue to the famous result of Carleson–Chang, namely that for $\varOmega = B_R(0)$ the above supremum is attained also in the limiting case $\alpha =\alpha _N$.