Positive solutions for a $2n$th-order $p$-Laplacian boundary value problem involving all even derivatives

Jiafa Xu, Zhongli Wei, Youzheng Ding

Abstract


In this paper, we investigate the existence and multiplicity
of positive solutions for the following
$2n$th-order $p$-Laplacian boundary value problem
$$
\cases
-(((-1)^{n-1}x^{(2n-1)})^{p-1})'\\
=f(t,x,-x^{\prime\prime},\ldots,(-1)^{n-1}x^{(2n-2)})
&\text{for } t\in [0,1],
\\
x^{(2i)}(0)=x^{(2i+1)}(1)=0 & \text{for } i=0,\ldots,n-1,
\endcases
$$
where $n\ge 1$ and $f\in C([0,1]\times \mathbb{R}_+^{n},
\mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$ depends on $x$ and all
derivatives of even orders. Based on a priori estimates achieved by
utilizing properties of concave functions and Jensen's integral
inequalities, we use fixed point index theory to establish our main
results. Moreover, our nonlinearity $f$ is allowed to grow
superlinearly and sublinearly.

Keywords


p-Laplacian equation; Jensen's inequality; positive solution; fixed point index; cone

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