Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem {u((4)) (x) + (k(1) + k(2))y ''(x) + k(1)k(2)y(x) =lambda f(x, y(x)), x is an element of [0, 1], y'(0) = y'(1) = y'''(0) = y'''(1) = 0, where k(1) and k(2) are constants, lambda > 0 is the bifurcation parameter, f is an element of C([0; 1] x R+; R), R+ := [0; infinity). 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 lambda near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

Automated summary

In this paper, we study the existence of positive solutions for a nonlinear fourth-order Neumann boundary value problem and prove the existence of a global branch of solutions emanating from infinity using global bifurcation theory.