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

  • J. F. de Oliveira
  • J. M. do Ó
  • B. RufEmail author


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 \(\alpha =\alpha _N\).


Trudinger–Moser inequality k-Hessian Extremal problem 

Mathematics Subject Classification

35J50 35J65 35J20 35J62 



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Authors and Affiliations

  1. 1.Department of MathematicsFederal University of PiauíTeresinaBrazil
  2. 2.Department of MathematicsBrasília UniversityBrasíliaBrazil
  3. 3.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly

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