On the positive solutions of semilinear equations $\Delta u+\lambda u-hu\sp p=0$ on the compact manifolds

In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations $\Delta u + \lambda u - h{u^p}= 0$ on compact Riemannian manifolds as well as on bounded smooth domains in ${R^n}$ with homogeneous Dirichlet or Neumann boundary conditions.