Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE (Formula presented.),t∈[0,T],where (Formula presented.) is a bounded function of constant sign, (Formula presented.) are the positive/negative part of a sign-changing weight a(t) and μ>0 is a real parameter. Depending on the sign of (Formula presented.) at infinity, we find existence/multiplicity of solutions for μ in a “small” interval near the value (Formula presented.).The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for (Formula presented.) and (Formula presented.) are given, as well.