Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

We prove the existence of positive periodic solutions for the second order nonlinear equation $u\mathrm{"}+a(x)g(u)=0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin’s coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.