Continuous dependence for $ p $-Laplace equations with varying operators

For the following Neumann problem in a ball $-\Delta_p u + u^{p-1} = u^{q-1}$ in B, $u > 0$, $u$ radial in $B$, partial derivative $u_{\partial\nu} = 0$ on $\partial B$, with $1 < p < q < \infty$, we prove continuous dependence on $p$, for radially nondecreasing solutions. As a byproduct, we obtain an existence result for nonconstant solutions in the case $p$ is an element of $(1,2)$ and $q$ larger than an explicit threshold.