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 subset of R and a(t) is a sign-changing weight function. We look for solutions u(t) of the differential equation such that u(t) is an element of I, satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for I = R-0(+) and g(u) similar to -u(-sigma), as well as the case of exponential nonlinearities, for I = R and g(u) similar to exp(u). The proofs are obtained by passing to an equivalent equation of the form x '' = f(x)(x')(2) + a(t).