    doi: 10.3934/cpaa.2021130
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Existence and multiplicity for Hamilton-Jacobi-Bellman equation

 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 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
 $s\neq0$
.
Citation: Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021130
##### References:
  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.  Google Scholar  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.  Google Scholar  R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.  doi: 10.1016/0362-546X(83)90081-0.  Google Scholar  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.  Google Scholar  C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.  doi: 10.1007/BF02414332.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar

show all references

##### References:
  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.  Google Scholar  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.  Google Scholar  R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.  doi: 10.1016/0362-546X(83)90081-0.  Google Scholar  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.  Google Scholar  C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.  doi: 10.1007/BF02414332.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar
  Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51  Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147  Norihisa Ikoma. Multiplicity of radial and nonradial solutions to equations with fractional operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3501-3530. doi: 10.3934/cpaa.2020153  Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034  Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839  Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095  Jie Yang, Haibo Chen. Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3065-3092. doi: 10.3934/cpaa.2021096  Norimichi Hirano, A. M. Micheletti, A. Pistoia. Existence of sign changing solutions for some critical problems on $\mathbb R^N$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 143-164. doi: 10.3934/cpaa.2005.4.143  Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405  Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603  Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999  Feliz Minhós, João Fialho. Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities. Conference Publications, 2013, 2013 (special) : 555-564. doi: 10.3934/proc.2013.2013.555  Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773  Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681  Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020  Gonzalo Galiano, Sergey Shmarev, Julian Velasco. Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1479-1501. doi: 10.3934/dcds.2015.35.1479  Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911  Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080  Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166  E. N. Dancer, Sanjiban Santra. Existence and multiplicity of solutions for a weakly coupled radial system in a ball. Communications on Pure & Applied Analysis, 2008, 7 (4) : 787-793. doi: 10.3934/cpaa.2008.7.787

2020 Impact Factor: 1.916

## Tools

Article outline

Figures and Tables