ON THE POSITIVE SOLUTIONS OF SEMILINEAR EQUATIONS DELTA-U+LAMBDA-U+HU(P)=0 ON COMPACT MANIFOLDS .2.

In this paper, we study the existence and non-existence of the positive solutions of the semilinear equation DELTA-u + lambda-u + hu(p) = 0 on compact Riemannian manifolds, where lambda > 0, p > 1 and h(x) satisfies assumptions (A-1) and (A-2) below. We prove that there exists a constant lambda-0, 0 < lambda-0 < infinity such that for any 0 < lambda < lambda-0, there exist two distinct solutions u1(lambda,x) < u2(lambda,x); for lambda = lambda-0, there exists a unique solution u(lambda-0,x); for lambda > lambda-0, there is no positive solution of the equation above.