AbstractWe show that for the prescribing scalar curvature problem on Sn (n = 3, 4), we can perturb (in an explicit way) any given positive continuous function in any neighborhood of any given point on Sn such that for the perturbed function there exist many solutions. The related critical exponent equations −Δu = K(x) u(n + 2)/(n −2) in Rn (n = 3, 4), with K(x) being asymptotically periodic in one of the variables, are also studied and infinitely many positive solutions (modulo translations by its periods) are obtained under some additional mild hypotheses on K(x). Some a priori estimates to solutions of the prescribing scalar curvature problem on S3 are as given.