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:; D. O'Regan, Department of Mathematics, National University of Ireland, Galway, Ireland, e-mail:; R. P. Agarwal, Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901, USA, e-mail:

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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