Green's Function and Existence Results for Solutions of Semipositone Nonlinear Euler–Bernoulli Beam Equations with Neumann Boundary Conditions

In this paper, we are concerned with the existence and multiplicity of positive solutions of the boundary value problem for the fourth-order semipositone nonlinear Euler–Bernoulli beam equation
$$ \begin{cases} y^{(4)}(x)+(\eta+\zeta)y''(x)+\eta\zeta y(x)=\lambda f(x,y(x)),& x\in[0,1],\\ y'(0)=y'(1)=y'''(0)=y'''(1)=0,& \end{cases} $$
where $\eta$ and $\zeta$ are constants, $\lambda>0$ is a parameter, and $f\in C([0,1]\times \mathbb{R}^+,\mathbb{R})$ is a function satisfying $f(x,y)\geq-\mathcal{X}$ for some positive constant $\mathcal{X}$; here $\mathbb{R}^+:=[0,\infty)$. The paper is concentrated on applications of the Green's function of the above problem to the derivation of the existence and multiplicity results for the positive solutions. One example is also given to demonstrate the results.