Global Structure of Positive Solutions for Semipositone Nonlinear Euler-Bernoulli Beam Equation with Neumann Boundary Conditions

Abstract

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem y (4)(x) + (k1 + k2)y 00(x) +  k1k2y(x) = λf(x, y(x)), x ∈ [0, 1], y 0 (0) = y 0 (1) = y 000(0) = y 000(1) = 0, where k1 and k2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × R +,  R), R + := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a  global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points,  solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.