Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 36, pp. 1-14.

Title: Positive solutions for a 2nth-order p-Laplacian  boundary
 value problem involving all derivatives
        
Authors: Youzheng Ding (Shandong Univ., Jinan, China)
         Jiafa Xu      (Shandong Univ., Jinan, China)
         Xiaoyan Zhang (Shandong Univ., Jinan, China)
        
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.