Introduction to semilinear elliptic boundary value problems

This paper provides a careful and accessible exposition of static bifurcation theory for degenerate boundary value problems for semilinear second-order elliptic differential operators. The purpose of this paper is twofold, The first purpose is to prove that the first eigenvalue of the linearized boundary value problem is simple and its associated eigenfunction is positive. The second purpose is to discuss the changes that occur in the structure of the solutions as a parameter varies near the first eigenvalue of the linearized problem.