We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.