Abstract By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system: d d t ( | u ˙ ( t ) | p − 2 u ˙ ( t ) ) − a ( t ) | u ( t ) | q − p u ( t ) + ∇ W ( t , u ( t ) ) = 0 , where 1 < p < ( q + 2 ) / 2 , q > 2 , t ∈ R , u ∈ R N , a ∈ C ( R , R ) and W ∈ C 1 ( R × R N , R ) are not periodic in t. MSC:34C37, 35A15, 37J45, 47J30.