The authors deal with boundary value problems of the type \[ (py')'+ p(t)q(t)f(t,y)=0, \;0<t<1, \qquad \lim_{t \to 0^+} p(t)y'(t)=0; \;y(1)=A\geq 0, \] where \(1/p\) is not necessarily summable and the nonlinear term \(f\) may change sign and is not necessarily Carathéodory. Moreover, note that for \(A=0\) the problem is singular.
The authors present two different approaches in order to obtain existence results. The first one is based on slight modifications on the classical upper and lower solutions technique. The second approach, valid for continuous nonlinearities \(f\), is based on certain growth conditions. Finally, the results are applied to a problem arising in the theory of shallow membrane caps.