Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

Abstract

We study the Neumann boundary value problem for the second order ODE

$$\begin{aligned} u^{\prime \prime } + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{aligned}$$

(1)

where $g \in {\mathcal {C}}^1({\mathbb {R}})$ is a bounded function of constant sign, $a^+,a^-: [0,T] \rightarrow {\mathbb {R}}^+$ are the positive/negative part of a sign-changing weight $a(t)$ and $\mu > 0$ is a real parameter. Depending on the sign of $g^{\prime }(u)$ at infinity, we find existence/multiplicity of solutions for $\mu$ in a “small” interval near the value

$$\begin{aligned} \mu _c = \frac{\int _0^T a^+(t) \, dt}{\int _0^T a^-(t) \, dt}\,. \end{aligned}$$

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 $\mu \rightarrow 0^+$ and $\mu \rightarrow +\infty$ are given, as well.