Reverse functional inequalities and their applications to nonlinear elliptic boundary value problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter $\lambda$ which have two branches of solutions $u\lambda(\lambda\geq 0)$ and $U\lambda(\lambda>0)$ of which the first is continuous at the origin and the second increases indefinitely as $\lambda\to 0$.