Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n-1|y|n. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γuγ-1 in two dimensions.