-
Previous Article
Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient
- DCDS-B Home
- This Issue
-
Next Article
Gaussian invariant measures and stationary solutions of 2D primitive equations
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 |
$ h $ |
$ p $ |
$ \begin{equation} \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. \end{equation} $ |
$ a(x) $ |
$ \Omega\subset \mathbb{R}^n(n\geq 2), $ |
$ h>1, $ |
$ 2< p<\infty, $ |
$ \Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right) $ |
$ p $ |
$ \lambda_\Omega $ |
$ p>n $ |
$ \lambda<\lambda_\Omega $ |
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. |
[1] |
E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209 |
[2] |
Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 |
[3] |
Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743 |
[4] |
Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335 |
[5] |
Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 |
[6] |
Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 |
[7] |
Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 |
[8] |
Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 |
[9] |
Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 |
[10] |
Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 |
[11] |
Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure and Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361 |
[12] |
Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 |
[13] |
Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 |
[14] |
Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587 |
[15] |
Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
[16] |
Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure and Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044 |
[17] |
Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure and Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1 |
[18] |
Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 |
[19] |
Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 |
[20] |
Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]