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

  • J. F. de Oliveira
  • J. M. do Ó
  • B. RufEmail author
Article
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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 \(\alpha =\alpha _N\).

Keywords

Trudinger–Moser inequality k-Hessian Extremal problem 

Mathematics Subject Classification

35J50 35J65 35J20 35J62 

Notes

References

  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  3. 3.
    Caffarelli, L., Nirenberg, L., Spruck, J.: Dirichlet problem for nonlinear second order elliptic equations III. Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113–127 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chou, K.S., Geng, D., Yan, S.S.: Critical dimension of a Hessian equation involving critical exponent and a related asymptotic result. J. Differ. Equ. 129, 79–110 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chou, K.S., Wang, X.-J.: Variational theory for Hessian equations. Commun. Pure Appl. Math. 54, 1029–1064 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clément, P., de Figueiredo, D.G., Mitidieri, E.: Quasilinear elliptic equations with critical exponents. Topol. Methods Nonlinear Anal. 7, 133–170 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55, 135–152 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Oliveira, J.F., do Ó, J.M.: Trudinger–Moser type inequalities for weighted Sobolev spaces involving fractional dimensions. Proc. Amer. Math. Soc. 142, 2813–2828 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kufner, A., Opic, B.: Hardy-type Inequalities, Pitman Research Notes in Mathematics, vol. 219. Longman Scientific and Technical, Harlow (1990)zbMATHGoogle Scholar
  11. 11.
    Labutin, D.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111, 1–49 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lions, P.L.: The concentration–compactness principle in the calculus of variations. The limit case, Part 1. Rev. Mat. Iberoam. 1, 145–201 (1985)CrossRefGoogle Scholar
  13. 13.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/1971)Google Scholar
  14. 14.
    Sánchez, J.: Bounded solutions of a k-Hessian equation in a ball. J. Differ. Equ. 261, 797–820 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sheng, W.M., Trudinger, N.S., Wang, X.-J.: The \(k\)-Yamabe problem. Surv. Differ. Geom. XVII, vol. 17, p. 427457 (2012)Google Scholar
  16. 16.
    Tian, G.-T., Wang, X.-J.: Moser–Trudinger type inequalities for the Hessian equation. J. Funct. Anal. 259, 1974–2002 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Trudinger, N.S., Wang, X.-J.: Hessian measures I. Topol. Methods Nonlinear Anal. 10, 225–239 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Trudinger, N.S., Wang, X.-J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, X.-J.: A class of fully nonlinear elliptic equations and related functionals. Indiana Univ. Math. J. 43, 25–54 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, X.-J.: The \(k\)-Hessian Equation, Lecture Notes in Mathematics, vol. 1977. Springer, Berlin (2009)Google Scholar
  22. 22.
    Wei, W.: Uniqueness theorems for negative radial solutions of k-Hessian equations in a ball. J. Differ. Equ. 261, 3756–3771 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wei, W.: Existence and multiplicity for negative solutions of k-Hessian equations. J. Differ. Equ. 263, 615–640 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

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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