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.