Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

We study the second order nonlinear differential equation $u'' + a(t) g(u) = 0$, where $g$ is a continuously differentiable function of constant sign defined on an open interval $I\subseteq {\mathbb R}$ and $a(t)$ is a sign-changing weight function. We look for solutions $u(t)$ of the differential equation such that $u(t)\in I,$ satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for $I = {\mathbb R}^+_0$ and $g(u) \sim - u^{-\sigma},$ as well as the case of exponential nonlinearities, for $I = {\mathbb R}$ and $g(u) \sim \exp (u).$ The proofs are obtained by passing to an equivalent equation of the form $x'' = f(x)(x')^2 + a(t).$