The Existence Of Solutions To A Class Of Semilinear Elliptic Equations

Semilinear elliptic equations arise in a variety of fields in natural science and play an important role in research of many subjects,such as physics,engineering,mechanics,life science etc.This paper is devoted to the study of the existence of solutions to a class of semilinear elliptic equations.Considering a particular nonlinear Euclidean scalar field equation as the auxiliary equation,we reduce an infinite-dimensional problem to a finite-dimensional one through the classical Lyapunov-Schmidt reduction method.Based on this,we study the linear projected problem,nonlinear projected problem and the reduced finite-dimensional problem respectively.We do not only prove that our problem possesses infinitely many positive solutions,but also show that these solutions are all nonradial.The point of our conclusion is that solutions to our problem are not necessarily radial even if both the solution to the auxiliary equation and the potential function in our problem are radial.