Aubry-Mather sets in semilinear asymmetric Duffing equations

In this paper, by introducing an appropriate action-angle variable transformation and adopting a new estimate method, we prove the existence of Aubry-Mather sets to a class of semilinear asymmetric Duffing equations $$ x''+\alpha x^{+}-\beta x^{-}+f(t,x)=0, $$ where \(x^{\pm }=\max \{\pm x,0\}\), α and β are positive constants satisfying $$ \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}=\frac{2}{\omega } $$ with \(\omega \in \mathbb{R}^{+} \), and \(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is a continuous function, 2π-periodic in the first variable and continuously differentiable in the second one, by virtue of a generalized version of Aubry-Mather theorem on cylinder with monotone twist assumption given by Pei. It should be pointed out that the perturbation term \(f(t,x)\) satisfying some suitable growth conditions, can be allowed to be either a bounded function or an unbounded function, which differs from many existing results in the literature. Moreover, some examples are provided to illustrate the validity of the proposed results.