Abstract: An existence theorem of positive strong solution for a Dirichlet problem with the critical Sobolev exponent was given by way of MountainPass Lemma and Strong Minimum Principle, which extended the corresponding results obtained by Brezis and Nirenberg to a threedimension space, the critical dimension space for the Dirichlet problem.

Key words: critical Sobolev exponent, positive strong solution, mountainpass lemma, strong minimum principle