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Bifurcation for nonlinear elliptic boundary value problems. III.

This paper is devoted to local static bifurcation theory for a class of \textit{degenerate} boundary value problems for nonlinear 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 \textit{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.