## Topological Methods in Nonlinear Analysis

### 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 $$\begin{cases} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\qquad t\in (0,1),\\ \displaystyle \\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\quad u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{cases} \quad i=1,\ldots,n.$$ Our analysis is based on critical point theory.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 42, Number 1 (2013), 105-118.

Dates
First available in Project Euclid: 21 April 2016

https://projecteuclid.org/euclid.tmna/1461247295

Mathematical Reviews number (MathSciNet)
MR3155617

Zentralblatt MATH identifier
1292.34018

#### Citation

Graef, John R.; Heidarkhani, Shapour; Kong, Lingju. Infinitely many solutions for systems of multi-point boundary value problems using variational methods. Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 105--118. https://projecteuclid.org/euclid.tmna/1461247295

#### References

• G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities , Bound. Value Probl., 2009 (2009), 1–20 \ref\key 2
• G. Bonanno and B. Di Bella, Infinitely many solutions for a fourth-order elastic beam equation, , NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357–368 \ref\key 3
• G. Bonanno and G. D'Agu\`i, A Neumann boundary value problem for the Sturm–Liouville equation , Appl. Math. Comput., 208 (2009), 318–327 \ref\key 4
• P. Candito and R. Livrea, Infinitely many solutions for a nonlinear Navier boundary value problem involving the $p$-biharmonic , Stud. Univ. Babes–Bolyai Math., 55 (2010), 41–51 \ref\key 5
• Z. Du, W. Ge and M. Zhou, Singular perturbations for third-order nonlinear multi-point boundary value problems , J. Differential Equations, 218 (2005), 69–90 \ref\key 6
• Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems , Electron. J. Qual. Theory Differ. Equ. (2009, 17 ). electronic \ref\key 7
• Z. Du, W. Liu and X. Lin, Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations , J. Math. Anal. Appl., 335 (2007), 1207–1218 \ref\key 8
• P.W. Eloe and B. Ahmad, Positive solutions of a nonlinear $n$-th order boundary value problem with nonlocal conditions , Appl. Math. Lett., 18 (2005), 521–527 \ref\key 9
• P.W. Eloe and J. Henderson, Uniqueness implies existence and uniqueness conditions for a class of $(k+j)$-point boundary value problems for $n$th order differential equations , Math. Nachr., 284 (2011), 229–239 \ref\key 10
• J.R. Graef, S. Heidarkhani and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems , Appl. Anal., 90 (2011), 1909–1925 \ref\key 11
• J.R. Graef, L. Kong and Q. Kong, Higher order multi-point boundary value problems , Math. Nachr., 284 (2011), 39–52 \ref\key 12
• J.R. Graef and B. Yang, Multiple positive solutions to a three point third order boundary value problem , Discrete Contin. Dyn. Syst. Suppl. (2005), 337–344 \ref\key 13
• J. Henderson, Existence and uniqueness of solutions of $(k+2)$-point nonlocal boundary value problems for ordinary differential equations , Nonlinear Anal., 74 (2011), 2576–2584 \ref\key 14
• J. Henderson, B. Karna and C.C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations , Proc. Amer. Math. Soc., 133 (2005), 1365–1369 \ref\key 15
• R. Ma, Existence of positive solutions for superlinear m-point boundary value problems , Proc. Edinburgh Math. Soc., 46 (2003), 279–292 \ref\key 16
• R. Ma and D. O'Regan, Solvability of singular second order $m$-point boundary value problems , J. Math. Anal. Appl., 301 (2007), 124–134 \ref\key 17
• B. Ricceri, A general variational principle and some of its applications , J. Comput. Appl. Math., 113 (2000), 401–410 \ref\key 18
• J.R.L. Webb, Optimal constants in a nonlocal boundary value problem , Nonlinear Anal., 63 (2005), 672–685 \ref\key 19
• J.R.L. Webb and G. Infante, Non-local boundary value problems of arbitrary order , J. Lond. Math. Soc. (2), 79 (2009), 238–258