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.
Bartsch, T., Degiovanni, M., Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains, <<ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI>>, 2006; 17 (1): 69-85. [doi:10.4171/RLM/454] [http://hdl.handle.net/10807/5134]
Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains
Degiovanni, Marco
2006
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.