### Infinitely many solutions for systems of multi-point boundary value problems using variational methods

#### Abstract

In this paper, we obtain the existence of infinitely many classical

solutions to the multi-point boundary value system

$$

\cases

-(\phi_{p_i}(u'_{i}))'=\lambda

F_{u_{i}}(x,u_{1},\ldots,u_{n}),\qquad t\in (0,1),\\

\noalign{\medskip}

\displaystyle

u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\quad u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j),

\endcases

\quad i=1,\ldots,n.

$$

Our analysis is based on critical point theory.

solutions to the multi-point boundary value system

$$

\cases

-(\phi_{p_i}(u'_{i}))'=\lambda

F_{u_{i}}(x,u_{1},\ldots,u_{n}),\qquad t\in (0,1),\\

\noalign{\medskip}

\displaystyle

u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\quad u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j),

\endcases

\quad i=1,\ldots,n.

$$

Our analysis is based on critical point theory.

#### Keywords

Infinitely many solutions; multi-point boundary value problems; multiplicity results; critical point theory

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