Open Access Research

Infinitely many solutions to superlinear second order m-point boundary value problems

Ruyun Ma1*, Chenghua Gao1 and Xiaoqiang Chen2

Full TextView PDF (311KB)

Author affiliations

1 Department of Mathematics, Northwest Normal University, Lanzhou, 730070, PR China

2 School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, PR China

For all author emails, please log on.

Citation and License

Boundary Value Problems 2011, 2011:14 doi:10.1186/1687-2770-2011-14

Published: 15 August 2011

Abstract

We consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M1">View MathML</a>

where:

(1) m ≥ 3, ηi ∈ (0, 1) and αi > 0 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M2">View MathML</a>;

(2) g : ℝ → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M3">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M4">View MathML</a>

(3) p : [0, 1] × ℝ2 → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M5">View MathML</a>

for some C > 0 and β ∈ (0, 1/2).

We obtain infinitely many solutions having specified nodal properties by the bifurcation techniques.

MSC(2000). 34B15, 58E05, 47J10

Keywords:
Nodal solutions; Second order equations; Multi-point boundary value problems; Bifurcation