Second Order Sturm-liouville Problems with Asymmetric, Superlinear Nonlinearities

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x), u′(x)), in (0, π), (1) c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, (2) where p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0, x ∈ [0, π], and ci0 + ci1 > 0, i = 0, 1. We suppose that f : [0, π] × R2 → R is continuous and there exist increasing functions ζl, ζu : [0,∞) → R, and a constant B, such that limt→∞ ζl(t) =∞ and ζl(ξ)ξ ≤ f(x, ξ, η) ≤ ζu(ξ)ξ, ξ ≥ 0, |f(x, ξ, η)| ≤ B|ξ|, ξ ≤ 0, for all (x, η) ∈ [0, π]×R (thus the nonlinearity in (1) is superlinear as u(x)→ ∞, but linearly bounded as u(x) → −∞). We obtain solutions of (1)–(2) having specified nodal properties.