Monotonicity of the Morse index of radial solutions of the H\'enon equation in dimension two

We consider the equation $$-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,x\in B,u=0\text{on }\mathrm{\partial }B,$$ 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$\alpha \to + \infty$.