In this work, we will find a family of small functions η y in the Kernel of an operator defined in the intersection of the Sobolev space H 2,q (S n ) with the orthogonal complement in H 1,2 (S n ) of the first eigenspace of the laplacian on S n , parameterized with a variable y belonging to a small ball contained in B n+1 .We will find L q estimates of these functions and we will use those estimates to find a subcritical solution to the scalar curvature problem on S n , and a solution2 (1+η y1 )•F y1 of a nonlinear elliptical problem related to that problem, where F y1 : S n → S n is a centered dilation.