Serrin-Type Overdetermined Problems for Hessian Quotient Equations and Hessian Quotient Curvature Equations

We consider overdetermined problems for Hessian quotient equations and Hessian quotient curvature equations, which are fully nonlinear elliptic equations. We establish Rellich–Pohozaev-type identities for Hessian quotient equations and Hessian quotient curvature equations. Based on these identities and the maximum principle for P functions, the symmetry of solutions can be proved in the Euclidean space. We also prove the related result for Hessian quotient equations in the hyperbolic space. Our results generalize the overdetermined problems for k-Hessian equations and k-curvature equations.