Positive solutions of singular four-point boundary value problem with $p$-Laplacian

Abstract

In this paper, we deal with the following singular four-point boundary value problem with $p$-Laplacian $$ \left\{\begin{aligned} &(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\ &u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0, \end{aligned}\right. $$ where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.