Quasilinear elliptic problems in RN involving oscillatory nonlinearities

In this paper we study an elliptic problem in RN which involves the p-Laplacian, p>N⩾2, and the nonlinear term has an oscillatory behavior. By means of a direct variational approach, we establish the existence of infinitely many homoclinic solutions whose W1,p(RN)-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively). The solutions have invariance properties with respect to certain subgroups of the orthogonal group O(N).