Non-Linear Second-Order Periodic Systems with Non-Smooth Potential

Proceedings Mathematical Sciences

Evgenia H Papageorgiou²  Nikolaos S, Papageorgiou¹ 

[1] $$;Department of Mathematics, National Technical University, Zografou Campus, Athens 0, Greece$$

关键词: Ordinary vector 𝑝-Laplacian;    non-smooth critical point theory;    locally Lipschitz function;    Clarke subdifferential;    non-smooth Palais–Smale condition;    homo-clinic solution;    problem at resonance;    Poincaré–Wirtinger inequality;    Landesman–Lazer type condition.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 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.

【 授权许可】

Unknown