Bifurcation for nonlinear elliptic boundary value problems. I (Q1924842)

This paper is devoted to local static bifurcation theory for a class of degenerate boundary value problems for nonlinear second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems. The purpose of this paper is twofold. The first is to prove that the first eigenvalue of the linearized boundary value problem is simple and the associated eigenfunction is positive. The second 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. More specifically, we characterize in detail the number of solutions under generic conditions on the quadratic and cubic terms.