Geodesic
Existence of solutions for a class of second-order $p$-Laplacian systems with impulsive effects
Applications of Mathematics, Tome 59 (2014) no. 5, pp. 543-570 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system \begin {gather} \frac {{\rm d}}{{\rm d}t}(|\dot {u}(t)|^{p-2}\dot {u}(t)) =\nabla F(t, u(t)),\quad \text {\rm a.e.}\ t\in [0,T],\nonumber \\ u(0)-u(T)=\dot {u}(0)-\dot {u}(T)=0,\nonumber \\ \Delta \dot {u}^i(t_{j})=\dot {u}^i(t_j^+)-\dot {u}^i(t_j^-)=I_{ij}(u^i(t_j)),\ i = 1, 2,\dots , N;\ j = 1, 2,\dots ,m.\nonumber \end {gather} By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.
The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system \begin {gather} \frac {{\rm d}}{{\rm d}t}(|\dot {u}(t)|^{p-2}\dot {u}(t)) =\nabla F(t, u(t)),\quad \text {\rm a.e.}\ t\in [0,T],\nonumber \\ u(0)-u(T)=\dot {u}(0)-\dot {u}(T)=0,\nonumber \\ \Delta \dot {u}^i(t_{j})=\dot {u}^i(t_j^+)-\dot {u}^i(t_j^-)=I_{ij}(u^i(t_j)),\ i = 1, 2,\dots , N;\ j = 1, 2,\dots ,m.\nonumber \end {gather} By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.
DOI : 10.1007/s10492-014-0071-5
Classification : 34B15, 34B37, 34C25, 58E30, 58E50
Keywords: second-order $p$-Laplacian Hamiltonian systems; impulsive effect; critical point theory 
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     title = {Existence of solutions for a class of second-order $p${-Laplacian} systems with impulsive effects},
     journal = {Applications of Mathematics},
     pages = {543--570},
     year = {2014},
     volume = {59},
     number = {5},
     doi = {10.1007/s10492-014-0071-5},
     mrnumber = {3255795},
     zbl = {06391450},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0071-5/}
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Chen, Peng; Tang, Xianhua. Existence of solutions for a class of second-order $p$-Laplacian systems with impulsive effects. Applications of Mathematics, Tome 59 (2014) no. 5, pp. 543-570. doi: 10.1007/s10492-014-0071-5

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