# Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

### Thomas Bartsch

Universität Giessen, Germany### Marco Degiovanni

Università Cattolica del Sacro Cuore, Brescia, Italy

## Abstract

Let $\Omega\subset\R^N$ be a ball or an annulus and $f:\R\to\R$ absolutely continuous, superlinear, subcritical, and such that $f(0)=0$. We prove that the least energy nodal solution of $-\Delta u= f(u)$, $u\in H^1_0(\Omega)$, is not radial. We also prove that Fu\v{c}ik eigenfunctions, i.~e.\ solutions $u\in H^1_0(\Omega)$ of $-\Delta u=\lambda u^+-\mu u^-$, with eigenvalue $(\lambda,\mu)$ on the first nontrivial curve of the Fu\v{c}ik spectrum, are not radial. A related result holds for asymmetric weighted eigenvalue problems.\\ An essential ingredient is a quadratic form generalizing the Hessian of the energy functional $J\in C^1(H^1_0(\Omega))$ at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.

## Cite this article

Thomas Bartsch, Marco Degiovanni, Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 1, pp. 69–85

DOI 10.4171/RLM/454