Qualitative Properties Of Solutions To Semilinear Elliptic Equations

The objective of this paper is to study a kind of semilinear partial differential equation —△u — λf(u) in bounded domain in R~N on which we use some basic tools and methods in the study of elliptic partial differential equations. We find out that the result depends heavily on the properties of f . Then we discuss the solution in the monotone case and the nonmonotone case. What's more , in the monotone case, according to the decay speed of f(t)-at , we make the exploration and get some good results. Finally, we point out some questions which haven't been conquered.