The Neumann problem for the generalized Hénon equation. Local analysis

For the boundary value problem $$\{\begin{array}{rcll}-{\mathrm{\Delta }}_{p}u+u^{p-1}& =& \mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{u}^{q-1}& in\mathrm{\Omega },\\ \frac{{\displaystyle \mathrm{\partial }u}}{{\displaystyle \mathrm{\partial }\mathbf{n}}}& =& 0& on\mathrm{\partial }\mathrm{\Omega },\end{array}$$ in the unit ball $\mathrm{\Omega }$, we investigate the properties of the positive radial solution. It is known, that for $1<p<n$, $\frac{(n-1)p}{n-p}<q<\frac{np}{n-p}$ and sufficiently large $\alpha$ this solution does not provide global minimum to the corresponding energy functional, see [M. Gazzini, E. Serra, 2008] for $p=2$ and [A.P. Shcheglova, 2018] in general case. Nevertheless, it is shown in [M. Gazzini, E. Serra, 2008] that for $n\ge 4$, $p=2$, $2<q<\frac{2n}{n-2}$ and sufficiently large $\alpha$ the radial solution is at least a local minimizer of the energy functional. We partially generalize this result. Namely, let $n\ge 4$ and let $p>2$ be sufficiently close to $2$. Then for all $p<q<\frac{np}{n-p}$, for sufficiently large $\alpha$ the second variation of the energy functional is positive. The same holds true for all $2<p<n$ if $q>p$ is sufficiently close to $p$.