Boundedness of solutions of nonlinear \(p\)-Laplacian (Q702547)

Subject of this paper is the nonlinear ordinary differential equation \((|x'|^{p-2}x')'+ g(x)= e(t)\) for the function \(x(t)\), where \(p\geq 2\), \(e(t)\) is a sufficiently smooth periodic function, and \(g(x)\) with \(xg(x)> 0\) satisfies some subquasilinear conditions for \(x\neq 0\). The author shows that under these assumptions all solutions of this equation are bounded. To this end, the equation is first written as a time-dependent planar Hamiltonian system for which action-angle variables are constructed. Subsequently, successive canonical transformations produce a system that, outside a sufficiently large disc, represents a perturbation of an integrable Hamiltonian system, whose Poincaré map is approximately a twist mapping. Application of Moser's twist theorem then yields the desired result.