Oscillation and nonoscillation theorems for second order nonlinear differential equations

New oscillation and nonoscillation theorems are obtained for the second order nonlinear differential equation $$ (|u'(t)|^{alpha -1} u'(t))' + p(t)|u(t)|^{alpha -1} u(t) = 0 $$ where $ p(t) in C [0, infty) $ and $ p(t) ge 0 $. Conditions only about the integrals of $ p(t) $ on every interval $ [2^n t_0, 2^{n+1} t_0] $ ($ n = 1, 2, ldots $) for some fixed $ t_0 >0 $ are used in the results.