# American Institute of Mathematical Sciences

May  2022, 27(5): 2701-2720. doi: 10.3934/dcdsb.2021155

## The eigenvalue problem for a class of degenerate operators related to the normalized $p$-Laplacian

 Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

Received  October 2020 Revised  March 2021 Published  May 2022 Early access  June 2021

Fund Project: This work was supported by National Natural Science Foundation of China (No. 11501292)

In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the
 $h$
degree homogeneous
 $p$
-Laplacian
 $$$\nonumber \left \{ \begin{array}{ll} {|Du|^{h-1}}\Delta_p^N u+ \lambda a(x)|u|^{h-1}u = 0, \quad\quad \rm{in}\quad \Omega, \\ u = 0, \quad\quad \quad \quad \rm{on} \quad\partial\Omega. \end{array}\right.$$$
Here
 $a(x)$
is a positive continuous bounded function in the closure of
 $\Omega\subset \mathbb{R}^n(n\geq 2),$
 $h>1,$
 $2< p<\infty,$
and
 $\Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right)$
is the normalized version of the
 $p$
-Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue
 $\lambda_\Omega$
, which is positive and has a corresponding positive eigenfunction for
 $p>n$
. The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter
 $\lambda<\lambda_\Omega$
, we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.
Citation: Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $p$-Laplacian. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2701-2720. doi: 10.3934/dcdsb.2021155
##### References:
 [1] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3. [2] M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 73 (1999), 276-285.  doi: 10.1007/s000130050399. [3] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105. [4] H. Berestycki, I.C. Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.  doi: 10.1016/j.matpur.2014.10.012. [5] I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations, 11 (2006), 91-119. [6] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366.  doi: 10.3934/cpaa.2007.6.335. [7] J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187-206.  doi: 10.1016/j.anihpc.2004.05.004. [8] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139.  doi: 10.1007/s005260000065. [10] K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361. [11] A. Elmoataz, M. Toutain and D. Tenbrinck., On the $p$-Laplacian and $\infty$-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793. [12] A. Elmoataz, X. Desquesnes and M. Toutain, On the game $p$-Laplacian on weighted graphs with applications in image processing and data clustering, European J. Appl. Math., 28 (2017), 922-948.  doi: 10.1017/S0956792517000122. [13] C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197. [14] H. Ishii, Viscosity solutions of non-linear partial differential equations, Sugaku Expositions, 9 (1996), 135-152. [15] H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z. [16] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368. [17] P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020. [18] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179. [19] B. Kawohl, S. Kröemer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differential Integral Equations, 27 (2014), 659-670. [20] M. Lewicka and J. J. Manfredi, Game theoretical methods in PDEs, Boll. Unione Mat. Ital., 7 (2014), 211-216.  doi: 10.1007/s40574-014-0011-z. [21] F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tug-of-war, Advanced Nonlinear Studies, 19 (2019), 89-112.  doi: 10.1515/ans-2018-2019. [22] Q. Liu and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal., 14 (2015), 167-184.  doi: 10.3934/cpaa.2015.14.167. [23] F. Liu, L. Tian and P. Zhao, A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian,, Nonlinear Analysis: TMA, 200 (2020), 112001, 15 pp. doi: 10.1016/j.na.2020.112001. [24] F. Liu and X. Yang, A weighted eigenvalue problem of the biased infinity Laplacian, Nonlinearity, 34 (2021), 1197-1237.  doi: 10.1088/1361-6544/abd85d. [25] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Eqns., 33 (2008), 1788-1817.  doi: 10.1080/03605300802289253. [26] P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as $p\rightarrow \infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.  doi: 10.1007/s13163-013-0124-4. [27] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073. [28] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Normale Sup. Pisa., 11 (2012), 215-241. [29] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var., 18 (2012), 81-90.  doi: 10.1051/cocv/2010046. [30] Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564.  doi: 10.1007/s00526-009-0298-2. [31] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1. [32] Y. Peres and S. Sheffield, Tug-of-war with noise: A game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048. [33] A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators,, C. R. Math. Acad. Sci. Paris, 342 (2006), 115–118. doi: 10.1016/j.crma.2005.11.003. [34] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.  doi: 10.1016/j.aim.2007.12.002.

show all references

##### References:
 [1] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3. [2] M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 73 (1999), 276-285.  doi: 10.1007/s000130050399. [3] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105. [4] H. Berestycki, I.C. Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.  doi: 10.1016/j.matpur.2014.10.012. [5] I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations, 11 (2006), 91-119. [6] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366.  doi: 10.3934/cpaa.2007.6.335. [7] J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187-206.  doi: 10.1016/j.anihpc.2004.05.004. [8] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139.  doi: 10.1007/s005260000065. [10] K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361. [11] A. Elmoataz, M. Toutain and D. Tenbrinck., On the $p$-Laplacian and $\infty$-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793. [12] A. Elmoataz, X. Desquesnes and M. Toutain, On the game $p$-Laplacian on weighted graphs with applications in image processing and data clustering, European J. Appl. Math., 28 (2017), 922-948.  doi: 10.1017/S0956792517000122. [13] C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197. [14] H. Ishii, Viscosity solutions of non-linear partial differential equations, Sugaku Expositions, 9 (1996), 135-152. [15] H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z. [16] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368. [17] P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020. [18] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179. [19] B. Kawohl, S. Kröemer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differential Integral Equations, 27 (2014), 659-670. [20] M. Lewicka and J. J. Manfredi, Game theoretical methods in PDEs, Boll. Unione Mat. Ital., 7 (2014), 211-216.  doi: 10.1007/s40574-014-0011-z. [21] F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tug-of-war, Advanced Nonlinear Studies, 19 (2019), 89-112.  doi: 10.1515/ans-2018-2019. [22] Q. Liu and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal., 14 (2015), 167-184.  doi: 10.3934/cpaa.2015.14.167. [23] F. Liu, L. Tian and P. Zhao, A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian,, Nonlinear Analysis: TMA, 200 (2020), 112001, 15 pp. doi: 10.1016/j.na.2020.112001. [24] F. Liu and X. Yang, A weighted eigenvalue problem of the biased infinity Laplacian, Nonlinearity, 34 (2021), 1197-1237.  doi: 10.1088/1361-6544/abd85d. [25] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Eqns., 33 (2008), 1788-1817.  doi: 10.1080/03605300802289253. [26] P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as $p\rightarrow \infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.  doi: 10.1007/s13163-013-0124-4. [27] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073. [28] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Normale Sup. Pisa., 11 (2012), 215-241. [29] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var., 18 (2012), 81-90.  doi: 10.1051/cocv/2010046. [30] Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564.  doi: 10.1007/s00526-009-0298-2. [31] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1. [32] Y. Peres and S. Sheffield, Tug-of-war with noise: A game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048. [33] A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators,, C. R. Math. Acad. Sci. Paris, 342 (2006), 115–118. doi: 10.1016/j.crma.2005.11.003. [34] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.  doi: 10.1016/j.aim.2007.12.002.
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