ON MULTIPLE SOLUTIONS OF A NONLINEAR DIRICHLET PROBLEM

Publisher Summary In the second-order case, if the nonlinearity of a nonlinear Dirichlet problem is replaced by a more general one, which need not be odd, and if λ 2 3 , where λ 2 and λ 2 are two eigenvalues of the elliptic operator, then the problem has more than three solutions. This chapter presents a short proof of an extension of Berger's theorem. Berger's approach is to apply the bifurcation theory and results on proper mappings to an abstract equation. The chapter presents a calculation of the Leray–Schauder indices of the zeros of a certain vector field and an application of a global theorem concerning the sum of these indices. How an abstract result due to Clark can be used to give a lower bound on the number of solutions of a class of boundary value problems is also described in the chapter. This bound is expressed in terms of the number of eigenvalues of L that are strictly less than λ.