Morse index of radial nodal solutions of H\'enon type equations in dimension two

We consider non-autonomous semilinear elliptic equations of the type \[ -\Delta u = |x|^{\alpha} f(u), \ \ x \in \Omega, \ \ u=0 \quad \text{on} \ \ \partial \Omega, \] where $\Omega \subset {\mathbb R}^2$ is either a ball or an annulus centered at the origin, $\alpha >0$ and $f: {\mathbb R}\ \rightarrow {\mathbb R}$ is $C^{1, \beta}$ on bounded sets of ${\mathbb R}$. We address the question of estimating the Morse index $m(u)$ of a sign changing radial solution $u$. We prove that $m(u) \geq 3$ for every $\alpha>0$ and that $m(u)\geq \alpha+ 3$ if $\alpha$ is even. If $f$ is superlinear the previous estimates become $m(u) \geq n(u)+2$ and $m(u) \geq \alpha+ n(u)+2$, respectively, where $n(u)$ denotes the number of nodal sets of $u$, i.e. of connected components of $\{ x\in \Omega; u(x) \neq 0\}$. Consequently, every least energy nodal solution $u_{\alpha}$ is not radially symmetric and $m(u_{\alpha}) \rightarrow + \infty$ as $\alpha \rightarrow + \infty$ along the sequence of even exponents $\alpha$.