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Existence of three solutions for impulsive multi-point boundary value problems. (English) Zbl 1368.34040

In this paper the existence of at least three nontrivial classical solutions is investigated for the following second order impulsive multi-point boundary value problem \[ \begin{cases} &-(\phi_{p_{i}}(u_i'))'=\lambda F_{u_{i}}(t,u_1,\ldots,u_n)+\mu G_{u_{i}}(t,u_1,\ldots,u_n),~~ t\in (0,1)\setminus Q, \\ & \Delta\phi_{p_{i}}(u_i'(t_j))=I_{ij}(u_i(t_j)), ~~ j=1,\ldots,m,\\ &u_i(0)=\sum_{k=1}^{\ell} a_k u_i(s_k), ~~ u_i(1)=\sum_{k=1}^{\ell} b_k u_i(s_k) \end{cases} \] for \(i=1,\ldots, n,\) where \(Q=\{t_1,\ldots,t_m\},\) \(p_i\in (1,\infty),\) \(\phi_{p_{i}}(x)=|x|^{p_i-2} x\) for \(i=1,\ldots, n,\) \(\lambda>0,\) \(\mu\geq 0\) are parameters, \(m, n, \ell\in \mathbb N,\) \(0<t_0<t_1<t_2<\ldots<t_m<t_{m+1}=1,\) \(0<s_1\leq s_2\leq \ldots\leq s_{\ell}<1,\) \(t_j\neq s_k, j=1,\ldots,m,~ k=1,\ldots,\ell\) and \(F, G: [0,1]\times {\mathbb R}^n\to \mathbb R\) are given functions.
Variational methods and critical point theory are used. An example illustrating the results is also presented.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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