Bounds on the number of solutions for elliptic equations with polynomial nonlinearities

We consider semilinear elliptic Neumann boundary value problems with polyno mial nonlinearities. Suppose that the degree n of the polynomial is odd and that the coefficient a(n) of the highest order term is strictly positive ( such that the corresponding nonlinear operator is globally coercive); then, if the coefficients of the lower order terms are sufficiently small, the e quation has for any given forcing term at most n solutions. The proof uses a Lyapunov-Schmidt procedure to reduce the problem to a one dimensional equ ation; using estimates on the lower order terms it is then shown that the o ne dimensional equation has at most n solutions. (C) 1999 Academic Press.