Existence of Positive Solutions for Higher Order p-Laplacian Boundary Value Problems

Abstract

In this paper, we consider a higher order p-Laplacian boundary value problem $$\begin{aligned} (-1)^n[\phi _{p}(u^{(2n-2)}+k^2u^{(2n-4)})]''=f(t,u), ~~0\le t\le 1,\\ u^{(2i)}(0)=0=u^{(2i)}(1), ~~ 0\le i \le n-1, \end{aligned}$$ where $$n\ge 1$$ and $$k\in (0, \frac{\pi }{2})$$ is a constant. By applying fixed point index theory, we derive sufficient conditions for the existence of positive solutions to the boundary value problem.