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.