Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Omega be a ball or an annulus in R^N and f absolutely continuous, superlinear, subcritical, and such that f(0)=0. We prove that the least energy nodal solution of -Delta u= f(u) is not radial. We also prove that Fucik eigenfunctions on the first nontrivial curve of the Fucik 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 at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.