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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 $$\frac{d}{dt}(\Phi_p(\dot{x}(t)))+\nabla F(t,x(t))=0,\qquad t \in [0,T],$$ $$ -\Delta \Phi_p(\dot{x}(t_i))=\nabla I_i(x(t_i)), \qquad i=1,2,\dots,l,$$ $$x(0)=x(T)=0$$ by using variational methods, where $T>0$, $x \in \mathbb{R}^N$, $p>1$, $0= t_0<t_1<\ldots<t_l<t_{l+1}=T$, $\Delta \Phi_p(\dot{x}(t_i))=\Phi_p(\dot{x}(t_i^+))-\Phi_p(\dot{x}(t_i^-))$, $\dot{x}(t_i^+)$ and $\dot{x}(t_i^-)$ denote, respectively, the right and left limits of $\dot{x}(t)$ at $t =t_i$, $\Phi_p(x)=|x|^{p-2} x$, $\nabla F(t,x)=\frac{\partial}{\partial x}F(t,x)$, $\nabla I_i(x)=\left( \frac{\partial I_i}{\partial x_1},\dots, \frac{\partial I_i}{\partial x_N} \right)$, $\nabla F(t,0) \not \equiv 0$, and $\nabla I_i \in C((\mathbb{R^+})^N, (\mathbb{R^+})^N)$. 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 {\it 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 [{\it 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 [{\it X. Lin} and {\it D. Jiang}, J. Math. Anal. Appl. 321, 501--514 (2006; Zbl 1103.34015)] by using the fixed point index in cones.

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
Full Text: DOI
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