# American Institute of Mathematical Sciences

November 2018, 17(6): 2309-2328. doi: 10.3934/cpaa.2018110

## Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian

 School of Mathematical Sciences, Shanghai Jiao Tong University, 200240, Shanghai, China

* Corresponding author

Received  May 2017 Revised  January 2018 Published  June 2018

Fund Project: The authors are supported partially by NSFC grant No. 11271253 and No. 11771285

In this paper, we investigate the Moser-Trudinger inequality when it involves a Finsler-Laplacian operator that is associated with functionals containing $F^2(\nabla u)$. Here $F$ is convex and homogeneous of degree 1, and its polar $F^o$ represents a Finsler metric on $\mathbb{R}^n$. We obtain an existence result on the extremal functions for this sharp geometric inequality.

Citation: Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
##### References:
 [1] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 275-293. [2] L. Carleson and S-Y. A. Chang, One the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127. [3] E. Dibenedetto, C1, α local regularity of weak solutions of degenerate elliptic equations, Nonliear Anal., 7 (1983), 827-850. [4] M. Flucher, Extremal functions for the Trudinger-Moser inequality in two dimensions, Comment. Math. Helv., 67 (1992), 471-497. [5] V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253. [6] J. Serrin, Local behavior of solutions of qusai-linear equations, Acta Math., 111 (1964), 247-302. [7] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential Theory of Degenerate Elliptic Equations, Oxford University Press, New York, 1993. [8] K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. [9] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. [10] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. [11] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Scet., 119 (1991), 125-136. [12] S. Pohozaev, The sobolev embedding in the special case pl = n, in Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach, Mathmatic sections, Mosco. Energet. inst., (1965), 158–170. [13] P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J. Math. Mech., 51 (1984), 126-150. [14] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. [15] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817. [16] G. F. Wang and C. Xia, A characterization of the wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 199 (2011), 99-115. [17] G. F. Wang and D. Ye, A hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. [18] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252 (2012), 1668-1770. [19] G. F. Wang and C. Xia, An optimal anisotropic Poincare inequality for convex domains, Pacific J. Math., 258 (2012), 305-325. [20] Y. Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension, J. Functi. Anal., 239 (2006), 100-126. [21] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.

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##### References:
 [1] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 275-293. [2] L. Carleson and S-Y. A. Chang, One the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127. [3] E. Dibenedetto, C1, α local regularity of weak solutions of degenerate elliptic equations, Nonliear Anal., 7 (1983), 827-850. [4] M. Flucher, Extremal functions for the Trudinger-Moser inequality in two dimensions, Comment. Math. Helv., 67 (1992), 471-497. [5] V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253. [6] J. Serrin, Local behavior of solutions of qusai-linear equations, Acta Math., 111 (1964), 247-302. [7] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential Theory of Degenerate Elliptic Equations, Oxford University Press, New York, 1993. [8] K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. [9] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. [10] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. [11] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Scet., 119 (1991), 125-136. [12] S. Pohozaev, The sobolev embedding in the special case pl = n, in Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach, Mathmatic sections, Mosco. Energet. inst., (1965), 158–170. [13] P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J. Math. Mech., 51 (1984), 126-150. [14] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. [15] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817. [16] G. F. Wang and C. Xia, A characterization of the wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 199 (2011), 99-115. [17] G. F. Wang and D. Ye, A hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. [18] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252 (2012), 1668-1770. [19] G. F. Wang and C. Xia, An optimal anisotropic Poincare inequality for convex domains, Pacific J. Math., 258 (2012), 305-325. [20] Y. Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension, J. Functi. Anal., 239 (2006), 100-126. [21] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.
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