# 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 & Applied Analysis, doi: 10.3934/cpaa.2021130
##### References:

show all references

##### References:
Bifurcation diagrams of Theorem 1.3
Bifurcation diagrams of Theorem 1.4
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