# 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.