Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ be a ball or an annulus and $f:\mathbb{R}\to \mathbb{R}$ absolutely continuous, superlinear, subcritical, and such that $f(0)=0$. We prove that the least energy nodal solution of $-\Delta u=f(u)$, $u\in H_0^1(\Omega )$, is not radial. We also prove that Fučik eigenfunctions, i.e. solutions $u\in H_0^1(\Omega )$ of $-\Delta u=\lambda u^{+}-\mu u^{-}$, with eigenvalue $(\lambda ,\mu )$ on the first nontrivial curve of the Fučik 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 $J\in C^1(H_0^1(\Omega ))$ at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.