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

#### 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.

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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