doi: 10.3934/cpaa.2022084
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Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators

Laboratoire de Mathématiques: Modélisation déterministe et aléatoire, Université de Sousse, Tunisia

Received  September 2021 Revised  February 2022 Early access May 2022

We establish a Liouville type result and also a characterization theorem for positive radial solutions of the Hardy-Hénon equation with the k-Hessian operators.

Citation: Kods Hassine. Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022084
References:
[1]

B. Barrios and A. Quaas, The sharp exponent in the study of the nonlocal Hénon equation in $ \mathbb{R}^n$: A Liouville theorem and an existence result, Calc. Var. Partial Differ. Equ., 59 (2020), 22 pp.

[2]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $ \mathbb{R}^n$ or $ \mathbb{R}^n_+$ through the method of moving planes, Commun. Partial Differ. Equ., 22 (1997), 1671-1690. 

[3]

M. F. Bidaut-Veron and L. Veron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. 

[4]

M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differ. Equ., 15 (2010), 1033-1082. 

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations Ⅲ: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[6]

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

[7]

D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. 

[8]

P. ClémentD. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.  doi: 10.12775/TMNA.1996.006.

[9]

J. DolbeaultM. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and euclidean spaces, Invent. Math., 206 (2016), 397-440. 

[10]

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

[11]

F. GladialiM. Grossi and S. L. N. Neves, Nonradial solutions for the Hénon equation in $ \mathbb{R}^n$, Adv. Math., 249 (2013), 1-36. 

[12]

J. Jacobsen, Global bifurcation problems associated with k-Hessian operators, Topol. Methods Nonlinear Anal., 14 (1999), 81-130.  doi: 10.12775/TMNA.1999.023.

[13]

T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. 

[14]

N. V. Krylov, Degenerate nonlinear elliptic equations, English transl. in Math. USSR-Sb., 48 (1984), 307-326. 

[15]

Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.

[16]

P. L. Lions, Two remarks on the Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275. 

[17]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.

[18]

Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, J. Differ. Equ., 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.

[19]

E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differ. Equ., 23 (2005), 301-326.  doi: 10.1007/s00526-004-0302-9.

[20]

D. SmetsJ. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725.

[21]

N. S. Trudinger and X. J. Wang, Hessian measures Ⅰ, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.  doi: 10.12775/TMNA.1997.030.

[22]

N. S. Trudinger and X. J. Wang, Hessian measures Ⅱ, Australian National University, Mathematics Research Report (ANU MRR) 035-97, 1997. doi: 10.12775/TMNA. 1997.030.

[23]

K. Tso, Remarks on critical exponents for Hessian operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 113-122.  doi: 10.1016/S0294-1449(16)30302-X.

[24]

J. Villavert, Classification of radial solutions to equations related to Caffarelli-Kohn-Nirenberg inequalities, Ann. Mate., 199 (2020), 299-315.  doi: 10.1007/s10231-019-00879-0.

[25]

X. J. Wang, Existence of multiple solutions to the equations of Monge-Ampère type, J. Differ. Equ., 100 (1992), 95-118.  doi: 10.1016/0022-0396(92)90127-9.

[26]

X. J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43 (1994), 25-54. 

[27]

W. WangK. Li and L. Hong, Nonexistence of positive solutions of $-\Delta u=K(x) u^p$ in $ \mathbb{R}^n$, Appl. Math. Lett., 18 (2005), 345-351.  doi: 10.1016/j.aml.2004.02.006.

[28]

P. WangZ. Dai and L. Cao, Radial symmetry and monotonicity for fractional Hénon equation in $ \mathbb{R}^n$, Complex Var. Elliptic Equ., 60 (2015), 1685-1695.  doi: 10.1080/17476933.2015.1041937.

show all references

References:
[1]

B. Barrios and A. Quaas, The sharp exponent in the study of the nonlocal Hénon equation in $ \mathbb{R}^n$: A Liouville theorem and an existence result, Calc. Var. Partial Differ. Equ., 59 (2020), 22 pp.

[2]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $ \mathbb{R}^n$ or $ \mathbb{R}^n_+$ through the method of moving planes, Commun. Partial Differ. Equ., 22 (1997), 1671-1690. 

[3]

M. F. Bidaut-Veron and L. Veron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. 

[4]

M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differ. Equ., 15 (2010), 1033-1082. 

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations Ⅲ: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[6]

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

[7]

D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. 

[8]

P. ClémentD. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.  doi: 10.12775/TMNA.1996.006.

[9]

J. DolbeaultM. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and euclidean spaces, Invent. Math., 206 (2016), 397-440. 

[10]

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

[11]

F. GladialiM. Grossi and S. L. N. Neves, Nonradial solutions for the Hénon equation in $ \mathbb{R}^n$, Adv. Math., 249 (2013), 1-36. 

[12]

J. Jacobsen, Global bifurcation problems associated with k-Hessian operators, Topol. Methods Nonlinear Anal., 14 (1999), 81-130.  doi: 10.12775/TMNA.1999.023.

[13]

T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. 

[14]

N. V. Krylov, Degenerate nonlinear elliptic equations, English transl. in Math. USSR-Sb., 48 (1984), 307-326. 

[15]

Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.

[16]

P. L. Lions, Two remarks on the Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275. 

[17]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.

[18]

Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, J. Differ. Equ., 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.

[19]

E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differ. Equ., 23 (2005), 301-326.  doi: 10.1007/s00526-004-0302-9.

[20]

D. SmetsJ. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725.

[21]

N. S. Trudinger and X. J. Wang, Hessian measures Ⅰ, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.  doi: 10.12775/TMNA.1997.030.

[22]

N. S. Trudinger and X. J. Wang, Hessian measures Ⅱ, Australian National University, Mathematics Research Report (ANU MRR) 035-97, 1997. doi: 10.12775/TMNA. 1997.030.

[23]

K. Tso, Remarks on critical exponents for Hessian operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 113-122.  doi: 10.1016/S0294-1449(16)30302-X.

[24]

J. Villavert, Classification of radial solutions to equations related to Caffarelli-Kohn-Nirenberg inequalities, Ann. Mate., 199 (2020), 299-315.  doi: 10.1007/s10231-019-00879-0.

[25]

X. J. Wang, Existence of multiple solutions to the equations of Monge-Ampère type, J. Differ. Equ., 100 (1992), 95-118.  doi: 10.1016/0022-0396(92)90127-9.

[26]

X. J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43 (1994), 25-54. 

[27]

W. WangK. Li and L. Hong, Nonexistence of positive solutions of $-\Delta u=K(x) u^p$ in $ \mathbb{R}^n$, Appl. Math. Lett., 18 (2005), 345-351.  doi: 10.1016/j.aml.2004.02.006.

[28]

P. WangZ. Dai and L. Cao, Radial symmetry and monotonicity for fractional Hénon equation in $ \mathbb{R}^n$, Complex Var. Elliptic Equ., 60 (2015), 1685-1695.  doi: 10.1080/17476933.2015.1041937.

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