A Singular Nonlinear Second-Order Neumann Boundary Value Problem with Positive Solutions

Feng Wang, Yujun Cui, Fang Zhang

Abstract

We study the existence and multiplicity of positive solutions for second-order Neumann boundary value problem −u′′ + a(t)u = h(t)f(t, u), t ∈ (0, 1), u′(0) = u′(1) = 0, where coefficient a(t) : [0, 1] → (−∞,+∞) is continuous and
$\max_{t∈[0,1]} a(t) > 0, h(t) may be singular at t = 0 and 1, moreover f(t, u) may also have singularity at u = 0. The first eigenvalue of the relevant linear problem and fixed point index theory are used in this study.