Solvability of a doubly singular boundary value problem arising in front propagation for reaction-diffusion equations

The paper deals with the solvability of the following doubly singular boundary value problem $$\begin{cases} \dot z = c g(u)-f(u) -\dfrac{h(u)}{z^\alpha}\\ z(0^+)=0, z(1^-)=0, \ z(u)>0 \text{ in } (0,1)\end{cases}$$ naturally arising in the study of the existence and properties of travelling waves for reaction-diffusion-convection equations governed by the $p-$Laplacian operator. Here $c,\alpha$ are real parameters, with $\alpha>0$, and $f,g,h$ are continuous functions in $[0,1]$, with $$h(0)=h(1), \quad h(u)>0 \text{ in } (0,1).$$