Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains
Thomas Bartsch, Marco Degiovanni · Rendiconti Lincei Matematica e Applicazioni · 2006
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 Fučik 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 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^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.