A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of on B1 (n⩾5) where μ and K>0 are smooth functions on B1. If K is very sub-harmonic at each critical point of K in B2/3 and the maximum of u in is comparable to its maximum over , then all positive solutions are uniformly bounded on . As an application, a priori estimate for solutions of equations defined on Sn is derived.