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

TL;DR

This paper investigates the Morse index of radial nodal solutions to Hénon type equations in two dimensions, establishing lower bounds related to the parameter lpha and the number of nodal sets, with implications for symmetry and stability.

Contribution

It provides new lower bounds for the Morse index of radial solutions to He9non equations, linking the index to lpha and the nodal structure, and shows non-radiality of least energy solutions.

Findings

Morse index lpha  for all lpha>0

Morse index lpha+3 if lpha is even

Morse index tends to infinity for least energy nodal solutions as lpha 

Abstract

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 $\alpha$.