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⊆R and a(t) is a sign-changing weight function. We look for solutions u(t) of the differential equation such that u(t)∈I, satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for I=R+0 and g(u)∼−u−σ, as well as the case of exponential nonlinearities, for I=R and g(u)∼exp(u) . The proofs are obtained by passing to an equivalent equation of the form x′′=f(x)(x′)2+a(t) .