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

摘要

We consider semilinear elliptic Neumann boundary value problems with polynomial 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 equation 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 equation; using estimates on the lower order terms it is then shown that the one dimensional equation has at most n solutions.