Electron. J. Diff. Equ., Vol. 2013 (2013), No. 36, pp. 1-14.

Positive solutions for a 2nth-order p-Laplacian boundary value problem involving all derivatives

Youzheng Ding, Jiafa Xu, Xiaoyan Zhang

Abstract:
In this work, we are mainly concerned with the positive solutions for the 2nth-order p-Laplacian boundary-value problem
$$\displaylines{ -(((-1)^{n-1}x^{(2n-1)})^{p-1})' =f(t,x,x',\ldots,(-1)^{n-1}x^{(2n-2)},(-1)^{n-1}x^{(2n-1)}),\cr x^{(2i)}(0)=x^{(2i+1)}(1)=0,\quad (i=0,1,\ldots,n-1), }$$
where $n\ge 1$ and $f\in C([0,1]\times \mathbb{R}_+^{2n}, \mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$. To overcome the difficulty resulting from all derivatives, we first convert the above problem into a boundary value problem for an associated second order integro-ordinary differential equation with p-Laplacian operator. Then, by virtue of the classic fixed point index theory, combined with a priori estimates of positive solutions, we establish some results on the existence and multiplicity of positive solutions for the above problem. Furthermore, our nonlinear term f is allowed to grow superlinearly and sublinearly.

Submitted September 10, 2012. Published January 30, 2013.
Math Subject Classifications: 34B18, 45J05, 47H1.
Key Words: Integro-ordinary differential equation; a priori estimate; index; fixed point; positive solution.

Show me the PDF file (253 KB), TEX file, and other files for this article.

Youzheng Ding
School of Mathematics, Shandong University
Jinan 250100, Shandong, China
email: dingyouzheng@139.com
Jiafa Xu
School of Mathematics, Shandong University
Jinan 250100, Shandong, China
email: xujiafa292@sina.com
Xiaoyan Zhang
School of Mathematics, Shandong University
Jinan 250100, Shandong, China
email: zxysd@sdu.edu.cn

Return to the EJDE web page