文摘
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 \(x'' = f(x)(x')^2 + a(t)\) .