Exact multiplicity and stability of solutions of second-order Neumann boundary value problem

The purpose of this paper is to establish the exact multiplicity and stability of solutions of the equation u″+g(x,u)=f(x)u″+g(x,u)=f(x) with the Neumann boundary value conditions u′(0)=u′(1)=0u′(0)=u′(1)=0. Exactly three ordered solutions are obtained by taking advantage of the anti-maximum principle combined with the methods of upper and lower solutions. Moreover, we obtain that one of three solutions is negative, while the other two are positive, the middle solution is unstable, and the remaining two are stable.