Boundedness in asymmetric oscillations at resonance (Q2492097)

The author considers the boundedness of solutions for the nonlinear equation \[ (\phi_p(x'))'+(p-1)[\alpha \phi_p(x^+)-\beta\phi_p(x^-)]=f(t), \tag{1} \] where \(\phi_p(u)=| u| ^{(p-2)}u\), \(p>1\) and \(\alpha\), \(\beta\) are positive constants satisfying the condition \(\alpha^{-1/p}+\beta^{-1/p}=\frac{2}{n}\). To prove the boundedness of solutions of equation (1), the author introduces the action and angle variable transformation. Under the transformation, equation (1) is transformed into a perturbation of an integrable Hamiltonian system outside of a large disc and then the Poincaré map of the transformed system is close to a twist map. The boundedness of solutions is guaranteed by Ortega's version of Moser's twist theorem.