On the Existence of Multiple Periodic Solutions for the Vector p-Laplacian via Critical Point Theory

Abstract

We study the vector p-Laplacian

$$(*)\quad \quad \quad \quad \quad \quad \quad \left\{ {_{u(0) = u(T),\quad u'(0) = u'(T),\quad 1 < p < \infty .}^{ - (|u'|^{p - 2} u')' = \nabla F(t,u)\quad \operatorname{a} .e.\quad t \in [0,T],} } \right.$$

We prove that there exists a sequence (u _n ) of solutions of (*) such that u _n is a critical point of ϕ and another sequence (u ^*_n ) of solutions of (*) such that u ^*_n is a local minimum point of ϕ, where ϕ is a functional defined below.