Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is not radial. We also prove that Fučik eigenfunctions, i. e. solutions u ∈ H 0 (Ω) of −∆u = λu − μu−, with eigenvalue (λ, μ) 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 ∈ C(H 0 (Ω)) at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.