Existence of solutions for a class of \(p\)-Laplacian systems with impulsive effects (Q439178)

The second-order \(p\)-Laplacian systems with impulsive effects NEWLINE\[NEWLINE\begin{cases} \frac{d}{dt}(|\dot u(t)|^{p-2}\dot u(t))=\nabla F(t,u(t))\, \text{for} \,\,\, \text{a.e}\,\,\, t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0,\\ \Delta \dot u^{i}(t_j)=I_{ij}(u^i(t_j)), i=1,2,\dots, N; j=1,2,\dots, m, \end{cases}NEWLINE\]NEWLINE is considered, where \(p>1\), \(T>0\), \(t_0=0<t_1<t_2<\dots<t_m<t_{m+1}=T\), \(I_{ij}: {\mathbb R} \to {\mathbb R}\), \(i=1,2,\dots, N\); \(j=1,2,\dots, m\), are continuous and \(F: [0,T]\times {\mathbb R}^N\to {\mathbb R}\) satisfies weak sublinear growth conditions.NEWLINENEWLINEExistence results are obtained by using the least action principle and the saddle point theorem, with or without impulsive effects improving some existing results in the literature.