On the existence of bounded positive solutions for a class of singular BVPs (Q943687)

The paper deals with the singular problem \[ -\left(u''(t)+k\frac{u'}{t}\right)=f_u(t,u(t)),\quad u'(0)=0=u(T), \leqno(1) \] where \(T>0\), \(k>1\) and \(f\) is the Carathéodory function on \([0,T]\times I\), \(I\) is an open interval containing \(0\). Under the assumption that the derivative \(f_u\) exists, is increasing and positive and that there exists a positive \(d\in I\) such that \[ \left| \int_0^T l^k f(l,d) \text{ d}l\right| <\infty,\quad \int_0^Tf_u(l,d)\text{ d}l <\frac{d}{T},\quad f_u(\cdot,d)\in L^2(0,T) \] it is proved that there exists a solution \(\bar u\in U\) of (1). Here, \(\bar u\) is the minimum of the energy functional associated with problem (1) and \(U\subset C^1[0,T]\) is defined by the conditions \[ 0\leq u(t)\leq d,\;u'(t)\leq 0 \;\text{on}\;[0,T],\;u(T)=0,\;t^ku'\in AC[0,T],\;u''\in L_{loc}^2(0,T). \]