JournalsrlmVol. 17 , No. 1DOI 10.4171/rlm/454

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

  • Thomas Bartsch

    Universität Giessen, Germany
  • Marco Degiovanni

    Università Cattolica del Sacro Cuore, Brescia, Italy
Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains cover

Abstract

Let ΩRN\Omega\subset\R^N be a ball or an annulus and f:RRf:\R\to\R absolutely continuous, superlinear, subcritical, and such that f(0)=0f(0)=0. We prove that the least energy nodal solution of Δu=f(u)-\Delta u= f(u), uH01(Ω)u\in H^1_0(\Omega), is not radial. We also prove that Fu\v{c}ik eigenfunctions, i.~e.\ solutions uH01(Ω)u\in H^1_0(\Omega) of Δu=λu+μu-\Delta u=\lambda u^+-\mu u^-, with eigenvalue (λ,μ)(\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 JC1(H01(Ω))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.