On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations

We consider the boundary value problem for nonlinear second-order differential equations of the form u″ + a(x)f(u) = 0, 0 < x < 1, u(0) = u(1) = 0. We establish the precise condition concerning the behavior of the ratio f(s)/s at infinity and zero for the existence of solutions with prescribed nodal properties. Then we derive the existence and the multiplicity of nodal solutions to the problem. Our argument is based on the shooting method together with the Strum's comparison theorem. The results obtained here can be applied to the study of radially symmetric solutions of the Dirichlet problem for semilinear elliptic equations in annular domains.