Non-linear second-order periodic systems with non-smooth potential
Evgenia H Papageorgiou, Nikolaos S Papageorgiou
TL;DR
The paper extends the theory of non-linear periodic systems driven by the vector $p$-Laplacian with locally Lipschitz non-smooth potentials using non-smooth critical point theory. It establishes existence, multiplicity, and homoclinic results under general growth conditions, and develops a scalar case with a generalized Landesman–Lazer type condition, including resonance analysis. The results rely on a careful variational framework, verification of a non-smooth PS-condition, and mountain-pass and saddle-point methods adapted to locally Lipschitz functionals. Overall, the work broadens the scope of hemivariational inequalities in periodic and homoclinic contexts for quasilinear systems. The methods and conditions provide new tools for studying non-smooth mechanics-inspired models governed by $p$-Laplacian dynamics.
Abstract
In this paper we study second order non-linear periodic systems driven by the ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the $p$-Laplacian. In the last section of the paper we examine the scalar \hbox{non-linear} and semilinear problem. Our approach uses a generalized Landesman--Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.