Multi-peak Solutions For The Prescribed Scalar Curvature Equation-?u=?1+?K?u^N-2/N-2

The equation -?u=(1+?K)u-N-2/N+2 is related to the problem of prescribing the scalar curvature on the standard sphere (SN,g0). In this paper, firstly we prove the existence of three peak solutions for the equation -?u=(1+?K)u-N-2/N+2,if K(x) has three critical points satisfying certain conditions. And then we give the existence of multi-peak solutions for the same equation, if K(x) has finite symmetry critical points satisfying certain conditions. We prove the existence of multi-peak solutions by Lyapunov-Schmidt finite dimension reduction method. Firstly, we give the function I?(u) related to equation, and transform the existence of multi-peak solutions into the existence of I?(u)'s critical points. Secondly, we define to construct the multi-peak solutions and it follows from the implicit function theorem that u is a positive critical point of I?(u) if and only if (?, y, ?, v) is a critical point of J?(u). Finally, we test there exists (?, y, ?, v) satisfying Lagrange multiplier theorem's four conditions and get the existence of multi-peak solutions, through related estimates, mountain pass theorem and degree theory knowledge.Prior to this, Cao.D, Noussair.E and Yan.S give the existence of two peak solutions in 2002 with article about the equation -?u=(1+?K)u-N-2/N+2. In this paper, the multi-peak solutions are more complicated, we prove the existence of multi-peak solutions, if K(x) has finite symmetrical critical points.