Continuous dependence for p-Laplace equations with varying operators

For the following Neumann problem in a ball $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u+u^{p-1}=u^{q-1}& \text{in }B,\\ u>0,u\text{ radial}& \text{in }B,\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0& \text{on }\mathrm{\partial }B,\end{array}$$ with $1<p<q<\mathrm{\infty }$, we prove continuous dependence on $p$, for radially nondecreasing solutions. As a byproduct, we obtain an existence result for nonconstant solutions in the case $p\in (1,2)$ and $q$ larger than an explicit threshold.