This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator: -△pu = λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.