Bounds on the Number of Solutions for Elliptic Equations with Polynomial Nonlinearities

We consider semilinear elliptic Neumann boundary value problems with polynomial nonlinearities. Suppose that the degreenof the polynomial is odd and that the coefficientanof 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 equation hasfor any given forcing termat mostnsolutions. The proof uses a Lyapunov–Schmidt procedure to reduce the problem to a one dimensional equation; using estimates on the lower order terms it is then shown that the one dimensional equation has at mostnsolutions.