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 $$\{\begin{array}{l}\dot{z}=cg(u)-f(u)-{\displaystyle \frac{h(u)}{z^{\alpha }}}\\ z(0^{+})=0,z(1^{-})=0,z(u)>0\text{ in }(0,1)\end{array}$$ 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 $$h(0)=h(1),h(u)>0\text{ in }(0,1)\mathrm{.}$$