Abstract: This paper is concerned with the existence of the weak positive solution of the following nonlinear Dirichlet problem on some conditions：－div（|∇u|p－2 ∇u）＋ a（x） u p－1 ＝ h（x） u q ＋ u p ∗ －1 ‚x ∈ R N ‚u ≥0‚u ≢0‚∫R Na（x）|u|p dx ＜＋∞．Where a： R N →R is continuous and nonnegative‚h： R N →R is some integrable function and2≤p＜N‚p2 ≤ N‚0＜q＜ p2 （p－1）N－p－1‚p ∗ ＝NpN－p ．Some results as p＝2or sub-critical exponent are generalized toP-Laplacian and critical exponent on weaker conditions‚and some results as a（x）＝0are generalized too．

Key words:  critical exponent, concentration compactness principle, Mountain Pass Geomtry, positive solution