Existence and multiplicity for Hamilton-Jacobi-Bellman equation

This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation\begin{document}$ \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*} $\end{document}where \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded regular domain with \begin{document}$ N\geq3 $\end{document} , \begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document} are general Hamilton-Jacobi-Bellman operators, \begin{document}$ \mu $\end{document} is a real parameter. By using bifurcation theory, we determine the range of parameter \begin{document}$ \mu $\end{document} of the above problem which has one or multiple constant sign solutions according to the behaviors of \begin{document}$ f $\end{document} at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document} , and whether \begin{document}$ f $\end{document} satisfies the signum condition \begin{document}$ f(s)s>0 $\end{document} for \begin{document}$ s\neq0 $\end{document} .