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$p$-Laplacian equation; periodic solution; critical point theory
We study the vector $p$-Laplacian \[ \left\rbrace \begin{array}{ll}-(| u^{\prime }| ^{p-2}u^{\prime })^{\prime }=\nabla F(t,u) \quad \text{a.e.}\hspace{5.0pt}t\in [0,T], u(0) =u(T),\quad u^{\prime }(0)=u^{\prime }(T),\quad 1<p<\infty . \end{array}\right. \qquad \mathrm{(*)}\] We prove that there exists a sequence $(u_n)$ of solutions of ($*$) such that $u_n$ is a critical point of $\varphi $ and another sequence $(u_n^{*}) $ of solutions of $(*)$ such that $u_n^{*}$ is a local minimum point of $\varphi $, where $\varphi $ is a functional defined below.
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