Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations

The existence of Aubry–Mather sets and infinitely many subharmonic solutions to the following p-Laplacian like nonlinear equation (p− 1)(φp(x)) + [αφp(x)− βφp(x)] + g(x) = h(t) is discussed, where φp(u) = |u|p−2u, p > 1, α, β are positive constants satisfying α − 1 p +β − 1 p = 2 n with n ∈ N, h is piece-wise two times differentiable and 2πp-periodic, g ∈ C1(R) is bounded, x± = max{±x, 0}, πp = 2π p sin(π/p) .