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

Abstract

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 $$x'' = f(x)(x')^2 + a(t)$$ .