We prove that positive solutions of the nonlinear Dirichlet problem where Ω is a ball or an annulus in and is a polynomial with nonnegative coefficients, have alternating power series representations at the origin. We also obtain several existence and nonuniqueness results for certain functions of the form with the critical exponent, where We give lower bounds for R in terms of the initial value u(0), where R is the radius of the ball Ω, an upper bound for u(r) for all 0<r<R, and an upper bound for|′(R)|. We also give several results for multiple solutions of this problem, showing that two such solutions must cross in (0,R), and give lower bounds for R in terms of the value of the crossover in this situation, as well as a nonexistence result.