Morse index of radial nodal solutions of Hénon type equations in dimension two

Abstract

We consider non-autonomous semilinear elliptic equations of the type \[ -Δu = |x|^α f(u), \ \ x \in Ω, \ \ u=0 \quad \text{on} \ \ \partial Ω, \] where $\Omega \subset {\mathbb{R}}^2$ is either a ball or an annulus centered at the origin, $\alpha >0$ and $f:\mathbb{R}\to \mathbb{R}$ 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 $m(u)\ge 3$ for every $\alpha >0$ and that $m(u)\ge \alpha +3$ if $\alpha$ is even. If $f$ is superlinear the previous estimates become $m(u)\ge n(u)+2$ and $m(u)\ge \alpha +n(u)+2$, respectively, where $n(u)$ denotes the number of nodal sets of $u$, i.e. of connected components of $\{x\in \Omega ;u(x)\ne 0\}$. Consequently, every least energy nodal solution $u_{\alpha }$ is not radially symmetric and $m(u_{\alpha })\to +\infty$ as $\alpha \to +\infty$ along the sequence of even exponents $\alpha$.

15 pages

References in corpus (1)

• A Reduction Method for Semilinear Elliptic Equations and Solutions Concentrating on Spheres

Cited by in corpus (1)

• On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's, Part II