A class of anisotropic expanding curvature flows

In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean \begin{document}$ R^{n+1} $\end{document} with speed \begin{document}$ u^\alpha\sigma_k^\beta $\end{document} firstly, where \begin{document}$ u $\end{document} is support function of the hypersurface, \begin{document}$ \alpha, \beta \in R^1 $\end{document} , and \begin{document}$ \beta>0 $\end{document} , \begin{document}$ \sigma_k $\end{document} is the \begin{document}$ k $\end{document} -th symmetric polynomial of the principal curvature radii of the hypersurface, \begin{document}$ k $\end{document} is an integer and \begin{document}$ 1\le k\le n $\end{document} . For \begin{document}$ \alpha\le1-k\beta $\end{document} , \begin{document}$ \beta>\frac{1}{k} $\end{document} we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for \begin{document}$ \alpha\le1-k\beta $\end{document} , \begin{document}$ \beta>\frac{1}{k} $\end{document} , we prove that the flow with the speed \begin{document}$ fu^\alpha\sigma_k^\beta $\end{document} exists for all time and converges smoothly after normalisation to a soliton which is a solution of \begin{document}$ fu^{\alpha-1}\sigma_k^{\beta} = c $\end{document} provided that \begin{document}$ f $\end{document} is a smooth positive function on \begin{document}$ S^n $\end{document} and satisfies that \begin{document}$ (\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}}) $\end{document} is positive definite. When \begin{document}$ \beta = 1 $\end{document} , our argument provides a proof to the well-known \begin{document}$ L_p $\end{document} Christoffel-Minkowski problem for the case \begin{document}$ p\ge k+1 $\end{document} where \begin{document}$ p = 2-\alpha $\end{document} , which is identify with Ivaki's recent result. Especially, we obtain the same result for \begin{document}$ k = n $\end{document} without any constraint on smooth positive function \begin{document}$ f $\end{document} . Finally, we also give a counterexample for the two anisotropic expanding flows when \begin{document}$ \alpha>1-k\beta $\end{document} .