Three-peak Solutions For The Fractional Prescribed Scalar Curvature Equation

Scalar curvature is an important and basic concept in mathematical physics,it is the simplest curvature invariant in Riemannian manifolds and Lagrangian density in general relativity.Therefore,the problem of scalar curvature has attracted widespread attention.The scalar curvature equation is a mathematical and physical model used to study the problem of scalar curvature on the unit sphere(SN,g0).In this paper,we mainly study the existence of the three-peak solution of the following fractional scalar curvature equation,where(?).In the third chapter of this article,We mainly use the Lyapunov-Schmidt finite dimensional reduction method to construct the three-peak solution(?)of the equation,and the energy of the three-peak solution is concentrated near the three different critical point of K(x).Firstly,we give the energy functional I?(u)corresponding to the fractional scalar curvature equation of this paper,let(?),transform finding critical points of I? into finding J?'s critical points,and then according to Lagrange's multiplier theorem,(?,y,?,v)is the equivalent condition of the critical point of J?.Finally,we constructe the three-peak solution of the equation by establishing a series of estimates,combining mountain pass theorem and the related content of degree theory.