# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021130
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## 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:

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##### References:
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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