Abstract
In this paper, suppose \(F: {\mathbb {R}}^{N} \rightarrow [0, +\infty )\) be a convex function of class \(C^{2}({\mathbb {R}}^{N} \backslash \{0\})\) which is even and positively homogeneous of degree 1. Firstly, we derive anisotropic Trudinger–Moser inequality with exact growth in \({\mathbb {R}}^N\), i.e., for any \(b>0\), there exists a constant \(C_{N, b}>0\) such that \(\int _{{\mathbb {R}}^{N}}\frac{\varPhi _N(\lambda |u|^{\frac{N}{N-1}})}{1+b|u|^{\frac{N}{N-1}}}dx \le C_{N, b}\Vert u\Vert _N^N, \quad \forall u\in W^{1, N}({\mathbb {R}}^{N}) \quad \text {with} \quad \int _{{\mathbb {R}}^N}F^{N}(\nabla u)dx \le 1, \) where \(\varPhi _N(t):=e^t-\sum _{k=0}^{N-2}\frac{t^k}{k!}\), \(\lambda \le \lambda _{N}=N^{\frac{N}{N-1}} \kappa _{N}^{\frac{1}{N-1}}\) and \(\kappa _{N}\) is the volume of a unit Wulff ball in \({\mathbb {R}}^N\). Moreover, this inequality fails if the power \(\frac{N}{N-1}\) is replaced by any \(p<\frac{N}{N-1}\). Secondly, we calculate the exact values of the supremums and give some results about nonexistence and existence of maximizers. Finally, we prove that anisotropic Trudinger–Moser inequality with the exact growth implies Trudinger–Moser inequality in \(W^{1, N}({\mathbb {R}}^N)\).
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References
Adachi, S., Tanaka, K.: Trudinger type inequalities in \({\mathbb{R}}^{N}\) and their best exponents. Proc. Am. Math. Soc. 128, 2051–2057 (2000)
Adams, D.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988)
Adimurthi, A., Sandeep, K.: A singular Moser–Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13, 585–603 (2007)
Adimurthi, A., Yang, Y.: An interpolation of Hardy inequality and Trudinger–Moser inequality. Int. Math. Res. Notices. 13, 2394–2426 (2010)
Alvino, A., Ferone, V., Trombetti, G., Lions, P.: Convex symmetrization and applications. Ann. Inst. H. Poincar\(\acute{e}\) Anal. Non Lin\(\acute{e}\)aire 14, 275–293 (1997)
Belloni, M., Ferone, V., Kawohl, B.: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z. Angew. Math. Phys. 54, 771–783 (2003)
Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. J. Hokkaido Math. 25, 537–566 (1996)
Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \({\mathbb{R}}^{2}\). Commun. Partial Differ. Equ. 17, 407–435 (1992)
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)
Chang, S.-Y.A., Yang, P.: The inequality of Moser and Trudinger and applications to conformal geometry. Commun. Pure Appl. Math. 56, 1135–1150 (2003)
Csató, G., Roy, P.: Extremal functions for the singular Moser–Trudinger inequality in 2 dimensions. Calc. Var. Partial Differ. Equ. 54, 2341–2366 (2015)
Csató, G., Roy, P.: Singular Moser–Trudinger inequality on simply connected domains. Commun. Partial Differ. Equ. 41, 838–847 (2016)
Csató, G., Nguyen, V.-H., Roy, P.: Extremals for the singular Moser–Trudinger inequality via \(n\)-harmonic transplantation. J. Differ. Equ. 270, 843–882 (2021)
de Figueiredo, D.-G., Miyagaki, O.-H., Ruf, B.: Elliptic equations in \({\mathbb{R}}^2\) with nonlinearities in the critical growth range. Calc. Var. Partial. Differ. Equ. 3, 139–153 (1995)
de Figueiredo, D.-G., do Ó, J.-M., Ruf, B.: Elliptic equations and systems with critical Trudinger–Moser nonlinearities. Discr. Contin. Dyn. Syst. 30, 455–476 (2011)
do Ó, J.-M.: N-Laplacian equations in \({\mathbb{R}}^{N}\) with critical growth. Abstr. Appl. Anal. 2, 301–315 (1997)
do Ó, J.-M., de Souza, M., de Medeiros, E., Severo, U.: An improvement for the Trudinger–Moser inequality and applications. J. Differ. Equ. 256, 1317–1349 (2014)
Ferone, V., Kawohl, B.: Remarks on a Finsler–Laplacian. Proc. Am. Math. Soc. 137, 247–253 (2009)
Flucher, M.: Extremal functions for Trudinger–Moser inequality in 2 dimensions. Comment. Math. Helv. 67, 471–497 (1992)
Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinburgh Sect. A 119, 125–136 (1991)
Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in \({\mathbb{R}}^N\). Math. Ann. 351(4), 781–804 (2011)
Ikoma, N., Ishiwata, M., Wadade, H.: Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints. Math. Ann. 373(1–2), 831–851 (2019)
Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger–Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS) 17(4), 819–835 (2015)
Ibrahim, S., Masmoudi, N., Nakanishi, K., Sani, F.: Sharp threshold nonlinearity for maximizing the Trudinger–Moser inequalities. J. Funct. Anal. 278, 108302 (2020)
Lam, N., Lu, G.: A new approach to sharp Moser–Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equ. 255, 298–325 (2013)
Lam, N., Lu, G.: Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. J. Geom. Anal. 24, 118–143 (2014)
Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities. Rev. Mat. Iberoam. 33, 1219–1246 (2017)
Lam, N., Lu, G., Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities. Adv. Math. 352, 1253–1298 (2019)
Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{N}\). Indiana Univ. Math. J. 57, 451–480 (2008)
Li, X., Yang, Y.: Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space. J. Differ. Equ. 264, 4901–4943 (2018)
Lin, K.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348, 2663–2671 (1996)
Lu, G., Tang, H.: Sharp singular Trudinger–Moser inequalities in Lorentz–Sobolev spaces. Adv. Nonlinear Stud. 16, 581–601 (2016)
Lu, G., Tang, H.: Sharp Moser–Trudinger inequalities on hyperbolic spaces with the exact growth condition. J. Geom. Anal. 26(2), 837–857 (2016)
Lu, G., Tang, H., Zhu, M.: Best constants for Adams inequalities with the exact growth condition in \({\mathbb{R}}^n\). Adv. Nonlinear Stud. 15(4), 763–788 (2015)
Malchiodi, A., Martinazzi, L.: Critical points of the Moser–Trudinger functional on a disk. J. Eur. Math. Soc. (JEMS) 16, 893–908 (2014)
Masmoudi, N., Sani, F.: Adams’ inequality with the exact growth condition in \({\mathbb{R}}^4\). Commun. Pure Appl. Math. 67(8), 1307–1335 (2014)
Masmoudi, N., Sani, F.: Trudinger–Moser inequalities with the exact growth condition in \({\mathbb{R}}^n\) and application. Commun. Partial Differ. Equ. 40, 1408–1440 (2015)
Mancini, G., Martinazzi, L.: The Moser–Trudinger inequality and its extremals on a disk via energy estimates. Calc. Var. Partial Differ. Equ. 20, 56–94 (2017)
Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)
Nguyen, V.-H.: The thresholds of the existence of maximizers for the critical sharp singular Moser-Trudinger inequality under constraints. Math. Ann. (2020). https://doi.org/10.1007/s00208-020-02010-8
Pohozaev, S.: The Sobolev embedding in the special case pl = n, in: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Sections, Moscov. Energet. Inst. Moscow. 158–170 (1965)
Peetre, J.: Espaces d’interpolation et theoreme de Soboleff. Ann. Inst. Fourier (Grenoble) 16, 279–317 (1966)
Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{2}\). J. Funct. Anal. 219, 340–367 (2005)
Ruf, B., Sani, F.: Sharp Adams-type inequalities in \({\mathbb{R}}^{n}\). Trans. Amer. Math. Soc. 365, 645–670 (2013)
Struwe, M.: Critical points of embeddings of \(H_{0}^{1, n}\) into Orlicz spaces. Ann. Inst. H. Poincar\(\acute{e}\) Anal. Non Lin\(\acute{e}\)aire 5, 425–464 (1988)
Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 697–718 (1976)
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)
Wang, G., Xia, C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 99, 99–115 (2011)
Wang, G., Xia, C.: Blow-up analysis of a Finsler–Liouville equation in two dimensions. J. Differ. Equ. 252, 1668–1700 (2012)
Xie, R., Gong, H.: A priori estimates and blow-up behavior for solutions of \(-Q_Nu = V e^u\) in bounded domain in \({\mathbb{R}}^{N}\). Sci. Chin. Math. 59, 479–492 (2016)
Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Dokl. 2, 746–749 (1961)
Zhou, C.-L., Zhou, C.-Q.: Moser–Trudinger inequality involving the anisotropic Dirichlet norm \((\int _{\varOmega }F^{N}(\nabla u)dx)^{\frac{1}{N}}\) on \(W_{0}^{1, N}(\varOmega )\). J. Funct. Anal. 276, 2901–2935 (2019)
Zhou, C.-L., Zhou, C.-Q.: On the anisotropic Moser-Trudinger inequality for unbounded domains in \({\mathbb{R}}^n\). Discrete Contin. Dyn. Syst. 40, 847–881 (2020)
Acknowledgements
This work was partially supported by CSC(No. 2019062400056). The author is grateful to Prof. Guofang Wang for his advice in this subject. The author would like to express his hearty thanks to the anonymous referee for his/her valuable comments and suggestions.
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Liu, Y. Anisotropic Trudinger–Moser inequalities associated with the exact growth in \({\mathbb {R}}^N\) and its maximizers. Math. Ann. 383, 921–941 (2022). https://doi.org/10.1007/s00208-021-02194-7
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DOI: https://doi.org/10.1007/s00208-021-02194-7