The existence of positive solutions for the singular two-point boundary value problem

Abstract

In this paper, we consider the following boundary value problem: $$ \begin{cases} ((-u'(t))^n)'=nt^{n-1}f(u(t)) & \text{for }0< t <1, \\ u'(0)=0,\quad u(1)=0, \end{cases} $$ where $n >1$. Using the fixed point theory on a cone and approximation technique, we obtain the existence of positive solutions in which $f$ may be singular at $u=0$ or $f$ may be sign-changing.