Monotonicity of the Morse index of radial solutions of the Hénon equation in dimension two

Abstract

We consider the equation \[ -Δu = |x|^α |u|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B, \] where $B\subset {\mathbb{R}}^2$ is the unit ball centered at the origin, $\alpha \ge 0$, $p>1$, and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets $n\ge 1$ of the solution $u_{\alpha ,n}$, we prove that the Morse index $m(u_{\alpha ,n})$ is monotone non-decreasing with respect to $\alpha$. Secondly, we provide a lower bound for the Morse indices $m(u_{\alpha ,n})$, which shows that $m(u_{\alpha ,n})\to +\infty$ as $\alpha \to +\infty$.