On the existence of multiple periodic solutions for the vector \(p\)-Laplacian via critical point theory. (English) Zbl 1099.34021

Summary: We study the vector \(p\)-Laplacian \[ \begin{cases} -(| u'| ^{p-2}u')'=\nabla F(t,u) \quad \text{a.e.}\;\;t\in [0,T], \\ u(0) =u(T),\quad u'(0)=u'(T),\quad 1<p<\infty . \end{cases} \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.


34B15 Nonlinear boundary value problems for ordinary differential equations
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