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August  2022, 21(8): 2799-2817. doi: 10.3934/cpaa.2022073

Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $

Lu Chen 1, , Guozhen Lu 2, and  Yansheng Shen 3,,

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
3. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author

Received  December 2021 Revised  February 2022 Published  August 2022 Early access  April 2022

Fund Project: The first author was partly supported by NSFC (No.11901031). The second author was partly supported by a grant from the Simons Foundation

In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on $ \mathbb{S}^n $. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $ \mathbb{S}^n $, we first employ the Mobius transform between $ \mathbb{S}^n $ and $ \mathbb{R}^n $, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $ \mathbb{S}^n $ is equivalent to some integral equation in $ \mathbb{R}^n $. Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on $ \mathbb{S}^n $. As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on $ \mathbb{S}^n $. Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on $ \mathbb{S}^n $.

Keywords: Uniqueness of nonnegative solutions, Lane-Emden equations, sharp constants, extremal functions, moving-plane method, GJMS operators on $ \mathbb{S}^n $.
Mathematics Subject Classification: 5J30, 46E35, 35B06, 35A02.
Citation: Lu Chen, Guozhen Lu, Yansheng Shen. Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2799-2817. doi: 10.3934/cpaa.2022073
References:
[1]

T. Aubin, Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296. 

[2]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.

[3]

T. Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 100 (1976), 149-173. 

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[5]

T. P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.  doi: 10.7146/math.scand.a-12120.

[6]

T. P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc., 347 (1995), 3671-3742.  doi: 10.7146/math.scand.a-12120.

[7]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York, 1957.

[8]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[9]

L. Chen, Z. Liu, G. Lu and C. Tao, Stein-Weiss inequalities with the fractional Poisson kernel, Rev. Mat. Iberoam., 36 (2020), 1289-1308.  doi: 10.4171/rmi/1167.

[10]

L. Chen, Z. Liu, G. Lu and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc., 370 (2018), 8429-8450.  doi: 10.1090/tran/7273.

[11]

L. Chen, G. Lu and C. Tao, Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals, Adv. Nonlinear Stud., 19 (2019), 475-494.  doi: 10.1515/ans-2018-2038.

[12]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[15]

Z. Djadli, E. Hebey and M. Ledoux, Paneitz type operators and applications, Duke Math. J., 104 (2000), 129-169.  doi: 10.1215/S0012-7094-00-10416-4.

[16]

C. Fefferman and C. R. Graham, Juhl's formulae for GJMS operators and Q-curvatures, J. Amer. Math. Soc., 26 (2013), 1191-1207.  doi: 10.1090/S0894-0347-2013-00765-1.

[17] C. Fefferman and C. R. Graham, The Ambient Metric, Princeton University Press, 2012. 
[18]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic boundary value problems, in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.

[19]

A. R. Gover and K. Hirachi, Conformally invariant powers of the Laplacian-a complete nonexistence theorem, J. Amer. Math. Soc., 17 (2004), 389-405.  doi: 10.1090/S0894-0347-04-00450-3.

[20]

C. R. Graham, Conformally invariant powers of the Laplacian, Ⅱ: nonexistence, J. London Math. Soc., 46 (1992), 566-576.  doi: 10.1112/jlms/s2-46.3.566.

[21]

C. Graham, R. Jenne, L. Mason and J. Sparling, Conformally invariant powers of the Laplacian. Ⅰ. Existence, J. London Math. Soc., 46 (1992), 557-565.  doi: 10.1112/jlms/s2-46.3.557.

[22]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Mathematical Analysis and Applications, Supplementary Studies, Academic Press, New York, 1981.

[23]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear ellipti equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[24]

F. Hang, On the higher order conformal covariant operators on the sphere, Commun. Contemp. Math., 9 (2007), 279-299.  doi: 10.1142/S0219199707002435.

[25]

F. Hang, X. Wang and X. Yan, An intergal equation in conformal geometry, Ann. Inst. H. Poincare Anal. Non Lineare., 26 (2009), 1-21.  doi: 10.1016/J.ANIHPC.2007.03.006.

[26]

A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, Geom. Funct. Anal., 23 (2013), 1278-1370.  doi: 10.1007/s00039-013-0232-9.

[27]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[28]

P. Lions, The concentration compactness principle in the calculus of variations. The limit case 1, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.

[29]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.

[30]

G. Lu, J. Wei and X. Xu, On conformally invariant equation $(-\Delta)^pu-K(x)u^{\frac{N+2p}{N-2p}} = 0$ and its generalizations, Ann. Mat. Pura Appl., 179 (2001), 309-329.  doi: 10.1007/BF02505961.

[31]

S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannianmanifolds, Symmetry Integr. Geom. Methods Appl., 4 (2008), 3 pp. doi: 10.3842/SIGMA.2008.036.

[32]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.  doi: 10.1080/00036819208840142.

[33]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin/Heidelberg, 1990.

[34]

E. Stein, Singular Integrals and Differentiablity Properties of Functions, Prineton Mathematical Series, Princeton University Press, Princeton, N.J. 1970.

[35]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[36]

C. Tao, Reversed Stein-Weiss inequalities with Poisson-type kernel and qualitative analysis of extremal functions, Adv. Nonlinear Stud., 21 (2021), 167-187.  doi: 10.1515/ans-2020-2112.

[37]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0(\Omega)$ into ${L^{\frac{{2N}}{{N - 4}}}}(\Omega )$, Differ. Integral Equ., 6 (1993), 259-276. 

[38]

X. J. Wang, Sharp constant in a Sobolev inequality, Nonlinear Anal., 20 (1993), 261-268.  doi: 10.1016/0362-546X(93)90162-L.

[39]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

show all references

References:
[1]

T. Aubin, Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296. 

[2]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.

[3]

T. Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 100 (1976), 149-173. 

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[5]

T. P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.  doi: 10.7146/math.scand.a-12120.

[6]

T. P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc., 347 (1995), 3671-3742.  doi: 10.7146/math.scand.a-12120.

[7]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York, 1957.

[8]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[9]

L. Chen, Z. Liu, G. Lu and C. Tao, Stein-Weiss inequalities with the fractional Poisson kernel, Rev. Mat. Iberoam., 36 (2020), 1289-1308.  doi: 10.4171/rmi/1167.

[10]

L. Chen, Z. Liu, G. Lu and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc., 370 (2018), 8429-8450.  doi: 10.1090/tran/7273.

[11]

L. Chen, G. Lu and C. Tao, Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals, Adv. Nonlinear Stud., 19 (2019), 475-494.  doi: 10.1515/ans-2018-2038.

[12]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[15]

Z. Djadli, E. Hebey and M. Ledoux, Paneitz type operators and applications, Duke Math. J., 104 (2000), 129-169.  doi: 10.1215/S0012-7094-00-10416-4.

[16]

C. Fefferman and C. R. Graham, Juhl's formulae for GJMS operators and Q-curvatures, J. Amer. Math. Soc., 26 (2013), 1191-1207.  doi: 10.1090/S0894-0347-2013-00765-1.

[17] C. Fefferman and C. R. Graham, The Ambient Metric, Princeton University Press, 2012. 
[18]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic boundary value problems, in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.

[19]

A. R. Gover and K. Hirachi, Conformally invariant powers of the Laplacian-a complete nonexistence theorem, J. Amer. Math. Soc., 17 (2004), 389-405.  doi: 10.1090/S0894-0347-04-00450-3.

[20]

C. R. Graham, Conformally invariant powers of the Laplacian, Ⅱ: nonexistence, J. London Math. Soc., 46 (1992), 566-576.  doi: 10.1112/jlms/s2-46.3.566.

[21]

C. Graham, R. Jenne, L. Mason and J. Sparling, Conformally invariant powers of the Laplacian. Ⅰ. Existence, J. London Math. Soc., 46 (1992), 557-565.  doi: 10.1112/jlms/s2-46.3.557.

[22]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Mathematical Analysis and Applications, Supplementary Studies, Academic Press, New York, 1981.

[23]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear ellipti equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[24]

F. Hang, On the higher order conformal covariant operators on the sphere, Commun. Contemp. Math., 9 (2007), 279-299.  doi: 10.1142/S0219199707002435.

[25]

F. Hang, X. Wang and X. Yan, An intergal equation in conformal geometry, Ann. Inst. H. Poincare Anal. Non Lineare., 26 (2009), 1-21.  doi: 10.1016/J.ANIHPC.2007.03.006.

[26]

A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, Geom. Funct. Anal., 23 (2013), 1278-1370.  doi: 10.1007/s00039-013-0232-9.

[27]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[28]

P. Lions, The concentration compactness principle in the calculus of variations. The limit case 1, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.

[29]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.

[30]

G. Lu, J. Wei and X. Xu, On conformally invariant equation $(-\Delta)^pu-K(x)u^{\frac{N+2p}{N-2p}} = 0$ and its generalizations, Ann. Mat. Pura Appl., 179 (2001), 309-329.  doi: 10.1007/BF02505961.

[31]

S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannianmanifolds, Symmetry Integr. Geom. Methods Appl., 4 (2008), 3 pp. doi: 10.3842/SIGMA.2008.036.

[32]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.  doi: 10.1080/00036819208840142.

[33]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin/Heidelberg, 1990.

[34]

E. Stein, Singular Integrals and Differentiablity Properties of Functions, Prineton Mathematical Series, Princeton University Press, Princeton, N.J. 1970.

[35]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[36]

C. Tao, Reversed Stein-Weiss inequalities with Poisson-type kernel and qualitative analysis of extremal functions, Adv. Nonlinear Stud., 21 (2021), 167-187.  doi: 10.1515/ans-2020-2112.

[37]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0(\Omega)$ into ${L^{\frac{{2N}}{{N - 4}}}}(\Omega )$, Differ. Integral Equ., 6 (1993), 259-276. 

[38]

X. J. Wang, Sharp constant in a Sobolev inequality, Nonlinear Anal., 20 (1993), 261-268.  doi: 10.1016/0362-546X(93)90162-L.

[39]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

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