Positive solutions for an $$n$$ n th-order quasilinear boundary value problem

Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the \(n\)th-order quasilinear boundary value problem $$\begin{aligned} {\left\{ \begin{array}{ll} -(\varphi (u^{(n-1)}))^\prime =f(t,u), \quad \text {a.e.}\ t\in [0,1],\\ u^{(i)}(0) = u^{(n-1)}(1)=0\ \quad (i=0, \ldots , n-2), \end{array}\right. } \end{aligned}$$ where \(n\geqslant 2\), \(\varphi : \mathbb R^+\rightarrow \mathbb R^+\) is either a convex or concave homeomorphism, and \(f\in C([0,1]\times \mathbb R^+,\mathbb R^+)(\mathbb R^+:=[0,\infty ))\). Based on a priori estimates achieved by utilizing Jensen’s inequalities for concave and convex functions, we use fixed point index theory to establish our main results.