Multiple solutions for systems of multi-point boundary value problems (Q2871596)

In this paper, the authors consider the multi-point boundary value problem NEWLINE\[NEWLINE\begin{aligned} &-\frac{d}{dx}(\left|\dot{u}_{i}\right|^{p_{i}-2}\dot{u}_{i})=\lambda F_{u_{i}}(x,u_{1},\dots,u_{n}),\;x\in (0,1),\\& u_{i}(0)=\sum^{m}_{j=1}a_{j}u_{i}(x_{j}),\;u_{i}(1)=\sum^{m}_{j=1}b_{j}u_{i}(x_{j}), \, i=1,\dots,n, \end{aligned}\eqno \;(1)NEWLINE\]NEWLINE where \(p_{i}>1\) for \(i=1,\dots,n\), \(\lambda\) is a positive parameter, \(m,n\geq 1\) are integers, \(a_{j}, b_{j}\in\mathbb R\) for \(j=1,\dots,m\), \(0<x_{1}< x_{2}<\dots < x_{m}< 1\) and \(F:[0,1]\times\mathbb R^{n}\longrightarrow\mathbb R\), \((x,t_{1},\dots,t_{n})\longmapsto F(x,t_{1},\dots,t_{n})\) is a continuous function, differentiable with respect to \((t_{1},\dots,t_{n})\) with continuous partial derivatives \(F_{t_{i}}=\frac{\partial F}{\partial t_{i}}\). Using critical point theory, they establish, under some suitable conditions, the existence of at least three solutions of the problem (1).