A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetric -Laplacian Differential Equations

For and , we propose a new estimate approach to study the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order asymmetric -Laplacian differential equations where and are two positive constants satisfying with , is a continuous function, -periodic in the first argument and continuously differentiable in the second one, , , and . Using the Aubry-Mather theorem given by Pei, we obtain the existence of Aubry-Mather sets and quasiperiodic solutions under some reasonable conditions. Particularly, the advantage of our approach is that it not only gives a simpler estimation procedure, but also weakens the smoothness assumption on the function in the existing literature.