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Existence results for second-order system with impulse effects via variational methods. (English) Zbl 1208.34032

This paper is devoted to the investigation of the existence of positive solutions to the second-order system with a $p$-Laplacian and impulses

$\frac{d}{dt}\left({{\Phi }}_{p}\left(\stackrel{˙}{x}\left(t\right)\right)\right)+\nabla F\left(t,x\left(t\right)\right)=0,\phantom{\rule{2.em}{0ex}}t\in \left[0,T\right],$
$-{\Delta }{{\Phi }}_{p}\left(\stackrel{˙}{x}\left({t}_{i}\right)\right)=\nabla {I}_{i}\left(x\left({t}_{i}\right)\right),\phantom{\rule{2.em}{0ex}}i=1,2,\cdots ,l,$
$x\left(0\right)=x\left(T\right)=0$

by using variational methods, where $T>0$, $x\in {ℝ}^{N}$, $p>1$, $0={t}_{0}<{t}_{1}<...<{t}_{l}<{t}_{l+1}=T$, ${\Delta }{{\Phi }}_{p}\left(\stackrel{˙}{x}\left({t}_{i}\right)\right)={{\Phi }}_{p}\left(\stackrel{˙}{x}\left({t}_{i}^{+}\right)\right)-{{\Phi }}_{p}\left(\stackrel{˙}{x}\left({t}_{i}^{-}\right)\right)$, $\stackrel{˙}{x}\left({t}_{i}^{+}\right)$ and $\stackrel{˙}{x}\left({t}_{i}^{-}\right)$ denote, respectively, the right and left limits of $\stackrel{˙}{x}\left(t\right)$ at $t={t}_{i}$, ${{\Phi }}_{p}\left(x\right)={|x|}^{p-2}x$, $\nabla F\left(t,x\right)=\frac{\partial }{\partial x}F\left(t,x\right)$, $\nabla {I}_{i}\left(x\right)=\left(\frac{\partial {I}_{i}}{\partial {x}_{1}},\cdots ,\frac{\partial {I}_{i}}{\partial {x}_{N}}\right)$, $\nabla F\left(t,0\right)¬\equiv 0$, and $\nabla {I}_{i}\in C\left({\left({ℝ}^{+}\right)}^{N},{\left({ℝ}^{+}\right)}^{N}\right)$.

The solutions of the problem are transferred, by considering an auxiliary problem, into the critical points of some functional, and the mountain pass theorem [see D. Guo, Nonlinear functional analysis, Shandong Science and Technology Press, Jinan (1985)] allows to prove the existence of at least one positive solution. Some results of [J. Simon, Lect. Notes Math. 665, 205–227 (1978; Zbl 0402.35017)], are also useful for the procedure. A particular case of this system has been studied in [X. Lin and D. Jiang, J. Math. Anal. Appl. 321, 501–514 (2006; Zbl 1103.34015)] by using the fixed point index in cones.

##### MSC:
 34B37 Boundary value problems for ODE with impulses 34B15 Nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 58E30 Variational principles on infinite-dimensional spaces
##### References:
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