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In this work we study the period function T of solutions to the conservative equationWe 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 u = u −1 in two dimensions.

Let M be a Cartan-Hadamard manifold with sectional curvature satisfying −b 2 ≤ K ≤ −a 2 < 0, b ≥ a > 0. Denote by ∂ ∞ M the asymptotic boundary of M and byM := M ∪ ∂ ∞ M the geometric compactification of M with the cone topology. We investigate here the following question: Given a finite number of points..,p k } extends continuously to p i , i = 1, ..., k, can one conclude that u ∈ C 0 M ? When dim M = 2, for Q belonging to a linearly convex space of quasi-linear elliptic operators S of the form IntroductionLet M be Cartan-Hadamard n−dimensional manifold (complete, connected, simply connected Riemannian manifold with non-positive sectional curvature). It is well-known that M can be compactified with the so called cone topology by adding a sphere at infinity, also called the asymptotic

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