Abstract: By using the method of upper and lower solutions, we discussed a class of elliptic equations with nonlinear terms of supercritical growth. We first proved the existence of solutions of the Dirichlet problem in the restricted region, and then proved that the local solutions in the restricted region could be extended to the whole region by combining a new  variational principle.

Key words: supercritical growth, variational method, upper and lower solutions, Carathéodory function