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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

d dt(Φ p (x ˙(t)))+F(t,x(t))=0,t[0,T],
-ΔΦ p (x ˙(t i ))=I i (x(t i )),i=1,2,,l,
x(0)=x(T)=0

by using variational methods, where T>0, x N , p>1, 0=t 0 <t 1 <...<t l <t l+1 =T, ΔΦ p (x ˙(t i ))=Φ p (x ˙(t i + ))-Φ p (x ˙(t i - )), x ˙(t i + ) and x ˙(t i - ) denote, respectively, the right and left limits of x ˙(t) at t=t i , Φ p (x)=|x| p-2 x, F(t,x)= xF(t,x), I i (x)=I i x 1 ,,I i x N , F(t,0)¬0, and I i C(( + ) N ,( + ) N ).

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:
34B37Boundary value problems for ODE with impulses
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
58E30Variational principles on infinite-dimensional spaces
References:
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