Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence, uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations. A necessary and sufficient condition for the existence of positive solutions to problems is given. We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations, then they are still valid when one perturbs the differential operator a little bit. As consequences, some uniqueness results of positive solutions under the domain perturbation are also obtained.