The Neumann problem for the generalized H\'enon equation. Local analysis

For the boundary value problem $$\left\{ \begin{array}{rcll} -\Delta_p u+u^{p-1}&=&|x|^{\alpha}u^{q-1}&\mbox{in }\Omega,\\ \frac{\displaystyle\partial u}{\displaystyle\partial{\bf n}}&=&0&\mbox{on }\partial \Omega, \end{array}\right.$$ in the unit ball $\Omega$, 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 $n\ge4$ 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 $p$.