**
APPLICATIONS OF MATHEMATICS, Vol. 50, No. 6, pp. 555-568, 2005
**

#
On the existence of multiple periodic solutions

for the vector $p$-Laplacian via

critical point theory

##
Haishen Lu, Donal O'Regan, Ravi P. Agarwal

* H. Lu*, Department of Applied Mathematics, Hohai University, Nanjing, 210098, China, e-mail: ` haishen2001@yahoo.com.cn`; * D. O'Regan*, Department of Mathematics, National University of Ireland, Galway, Ireland, e-mail: ` Donal.ORegan@nuigalway.ie`; R. P. Agarwal, Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901, USA, e-mail: ` agarwal@fit.edu`

**Abstract:** We study the vector $p$-Laplacian

\cases-(| u'| ^{p-2}u')'=\nabla F(t,u) \quad\text{a.e.}\enspace t\in[0,T],

u(0) =u(T),\quad u'(0)=u'(T),\quad1<p<\infty. \tag{$*$}

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.

**Keywords:** $p$-Laplacian equation, periodic solution, critical point theory

**Classification (MSC 2000):** 34B15

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