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

TL;DR

This paper studies the properties of positive radial solutions to a Neumann boundary value problem for a generalized Hénon equation in the unit ball, focusing on their local minimality of the energy functional under various parameter conditions.

Contribution

It extends previous results by showing that for certain parameter ranges, the second variation of the energy functional is positive, indicating local minimality of solutions.

Findings

For large lpha, solutions are at least local minimizers of the energy functional.

The second variation is positive for p close to 2 and q in a specific range.

The results generalize known cases for p=2 to p>2 close to 2.

Abstract

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 $p$.