A Class of inverse curvature type flows in Rn+1

In this paper, we consider an expanding flow of smooth, closed, (η,k)-convex hypersurfaces in Euclidean Rⁿ⁺¹ with speed uαρδσₖ-β/k(λ(η)), where u, ρ are the support function and radical function of the hypersurface, respectively, α,δ¹, β>0, k is an integer and 1 ≤ k ≤ n, η=Hg-h, the first Newton transformation of the second fundamental form h, λ(η) denote the eigenvalues of g⁻¹η. For α+δ+β≤ 1, we prove that the flow has a unique smooth and (η,k)-convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for α+δ+β< 1, we prove that the flow with the speed fuαρδσₖ-β/k(λ(η)) exists for all time, and converges smoothly after normalisation to a soliton, which is a solution of fuα-1ρδσₖ-β/k(λ(η))=γ, provided that f is a smooth positive function on Sⁿ and γ is a positive constant. What's more, we can use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.