Electron. J. Diff. Equ., Vol. 2009(2009), No. 154, pp. 1-9.

Positive solutions for third-order Sturm-Liouville boundary-value problems with p-Laplacian

Chengbo Zhai, Chunmei Guo

Abstract:
In this article, we consider the third-order Sturm-Liouville boundary value problem, with $p$-Laplacian,
$$\displaylines{ (\phi_p(u''(t)))'+f(t,u(t))=0, \quad t\in (0,1),\cr \alpha u(0)-\beta u'(0)=0,\quad \gamma u(1)+\delta u'(1)=0,\quad u''(0)=0, }$$
where $\phi_p(s)=|s|^{p-2}s$, $p>1$. By means of the Leggett-Williams fixed-point theorems, we prove the existence of multiple positive solutions. As an application, we give an example that illustrates our result.

Submitted September 1, 2008. Published November 28, 2009.
Math Subject Classifications: 34K10
Key Words: Positive solution; Sturm-Liouville boundary value problem; p-Laplacian operator; concave functional; fixed point.

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Chengbo Zhai
School of Mathematical Sciences, Shanxi University
Taiyuan 030006, Shanxi, China
email: cbzhai@sxu.edu.cn
Chunmei Guo
School of Mathematical Sciences, Shanxi University
Taiyuan 030006, Shanxi, China
email: guocm@sxu.edu.cn

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