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November  2021, 20(11): 3767-3793. doi: 10.3934/cpaa.2021130

Existence and multiplicity for Hamilton-Jacobi-Bellman equation

Bian-Xia Yang 1,, , Shanshan Gu 1, and  Guowei Dai 2,

1. 

College of Science, Northwest A & F University, Yangling, Shaanxi 712100, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author

Received  December 2020 Revised  June 2021 Published  November 2021 Early access  July 2021

Fund Project: The first author is supported by Natural Science Basic Research Program of Shaanxi (Program No.2020JQ-237)

This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation
$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $
where
$ \Omega\subset\mathbb{R}^N $
 is a bounded regular domain with
$ N\geq3 $
,
$ \mathcal{M}_\mathcal{C}^{\pm} $
 are general Hamilton-Jacobi-Bellman operators,
$ \mu $
 is a real parameter. By using bifurcation theory, we determine the range of parameter
$ \mu $
 of the above problem which has one or multiple constant sign solutions according to the behaviors of
$ f $
 at
$ 0 $
 and
$ \infty $
, and whether
$ f $
 satisfies the signum condition
$ f(s)s>0 $
 for
$ s\neq0 $
.
Keywords: Extremal operators, bifurcation, existence, multiplicity, constant sign solutions.
Mathematics Subject Classification: Primary: 35B32, 35B50, 35F21; Secondary: 35P30.
Citation: Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130
References:
[1]

S. Alarcón, L. Iturriaga and A. Quaas, Existence and multiplicity results for Pucci's operators involving nonlinearities with zeros, Calc. Var. Partial Differ. Equ., 45 (2012), 443-454.  doi: 10.1007/s00526-011-0465-0.

[2]

S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differ. Equ., 246 (2009), 2958-2987.  doi: 10.1016/j.jde.2008.10.026.

[3] R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957. 
[4]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287. doi: 10.5802/afst.1070.

[5]

J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. I. H. Poincaré-AN, 22 (2005), 187-206.  doi: 10.1016/j.anihpc.2004.05.004.

[6]

A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control., Translated from French, In: ''Studies in Mathematics and its Applications" 12, North-Holland Publishing Co., Amsterdam New York, 1982.

[7]

G. W. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.  doi: 10.3934/dcds.2016034.

[8]

G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differ. Equ., 55 (2016), 97pp. doi: 10.1007/s00526-016-1029-0.

[9]

G. W. Dai, Generalized limit theorem and bifurcation for problems with Pucci's operator, Topol. Methods Nonlinear Anal., 56 (2020), 229-261.  doi: 10.12775/TMNA.2020.012.

[10]

P. L. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (11) (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.

[11]

B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[12]

Y. X. Hua and X. H. Yu, Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation, J. Math. Anal. Appl., 405 (2013), 608-617.  doi: 10.1016/j.jmaa.2013.04.025.

[13]

H. J. Kappen, Optimal control theory and the linear Bellman Equation, Inference and Learning in Dynamic Models, (2011), 363–387. doi: https://doi.org/10.1017/CBO9780511984679.018.

[14]

P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.

[15]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Commun. Partial Differ. Equ., 8 (1983), 1101-1174.  doi: 10.1080/03605308308820297.

[16]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Commun. Partial Differ. Equ., 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301.

[17]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. III. Regularity of the optimal cost function, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, V (1983), 95–205. doi: https://doi.org/10.1017/CBO9780511984679.018.

[18]

P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.  doi: 10.1016/0362-546X(83)90081-0.

[19]

C. Pucci, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.  doi: 10.2307/2036253.

[20]

C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.  doi: 10.1007/BF02414332.

[21]

A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Differ. Integral Equ., 17 (2004), 481-494.

[22]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci's operator, Commun. Partial Differ. Equ., 31 (2006), 987-1003.  doi: 10.1080/03605300500394421.

[23]

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.

[24]

A. Quaas and A. Allendes, Multiplicity results for extremal operators through bifurcation, Discrete Contin. Dyn. Syst., 29 (2011), 51-65.  doi: 10.3934/dcds.2011.29.51.

[25]

X. H. Yu, Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros, Comm. Pure Appl. Math., 12 (2013), 451-459.  doi: 10.3934/cpaa.2013.12.451.

show all references

References:
[1]

S. Alarcón, L. Iturriaga and A. Quaas, Existence and multiplicity results for Pucci's operators involving nonlinearities with zeros, Calc. Var. Partial Differ. Equ., 45 (2012), 443-454.  doi: 10.1007/s00526-011-0465-0.

[2]

S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differ. Equ., 246 (2009), 2958-2987.  doi: 10.1016/j.jde.2008.10.026.

[3] R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957. 
[4]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287. doi: 10.5802/afst.1070.

[5]

J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. I. H. Poincaré-AN, 22 (2005), 187-206.  doi: 10.1016/j.anihpc.2004.05.004.

[6]

A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control., Translated from French, In: ''Studies in Mathematics and its Applications" 12, North-Holland Publishing Co., Amsterdam New York, 1982.

[7]

G. W. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.  doi: 10.3934/dcds.2016034.

[8]

G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differ. Equ., 55 (2016), 97pp. doi: 10.1007/s00526-016-1029-0.

[9]

G. W. Dai, Generalized limit theorem and bifurcation for problems with Pucci's operator, Topol. Methods Nonlinear Anal., 56 (2020), 229-261.  doi: 10.12775/TMNA.2020.012.

[10]

P. L. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (11) (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.

[11]

B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[12]

Y. X. Hua and X. H. Yu, Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation, J. Math. Anal. Appl., 405 (2013), 608-617.  doi: 10.1016/j.jmaa.2013.04.025.

[13]

H. J. Kappen, Optimal control theory and the linear Bellman Equation, Inference and Learning in Dynamic Models, (2011), 363–387. doi: https://doi.org/10.1017/CBO9780511984679.018.

[14]

P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.

[15]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Commun. Partial Differ. Equ., 8 (1983), 1101-1174.  doi: 10.1080/03605308308820297.

[16]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Commun. Partial Differ. Equ., 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301.

[17]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. III. Regularity of the optimal cost function, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, V (1983), 95–205. doi: https://doi.org/10.1017/CBO9780511984679.018.

[18]

P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.  doi: 10.1016/0362-546X(83)90081-0.

[19]

C. Pucci, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.  doi: 10.2307/2036253.

[20]

C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.  doi: 10.1007/BF02414332.

[21]

A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Differ. Integral Equ., 17 (2004), 481-494.

[22]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci's operator, Commun. Partial Differ. Equ., 31 (2006), 987-1003.  doi: 10.1080/03605300500394421.

[23]

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.

[24]

A. Quaas and A. Allendes, Multiplicity results for extremal operators through bifurcation, Discrete Contin. Dyn. Syst., 29 (2011), 51-65.  doi: 10.3934/dcds.2011.29.51.

[25]

X. H. Yu, Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros, Comm. Pure Appl. Math., 12 (2013), 451-459.  doi: 10.3934/cpaa.2013.12.451.

Figure 1.  Bifurcation diagrams of Theorem 1.3
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Figure 2.  Bifurcation diagrams of Theorem 1.4
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