Angela Pistoia

Universitá di Roma 1 (Italy)

On the existence of sign changing solutions for the Bahri-Coron problem

Let Ω be a bounded smooth domain in RN, N 3 and p = N+2 N-2. We are interested in existence and multiplicity of sign changing solutions to the slightly subcritical problem

(1)    - Δu  = |u|p-1- εu in Ω, u = 0 on ∂Ω,
and to the Bahri-Coron’s problem
(2)    - Δu = |u|p-1u in Ωε, u = 0 on ∂Ω ε,
when Ωε = Ω \B(0). In both cases ε is a small positive parameter. We prove that, problem (1) has at least N pairs of solutions which change sign exactly once. Moreover, the nodal surface of these solutions intersects the boundary of Ω, provided some suitable conditions are satisfied ([1]). When Ω is symmetric and contains the origin, we construct sign changing solutions to problems (1) ([3]) and (2) ([2]) with multiple blow up at the origin. These solutions have, as ε goes to zero, more and more annular-shaped nodal domains.
  1. T. Bartsch, A.M. Micheletti, A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equations (to appear).
  2. M. Musso, A. Pistoia, Sign changing solutions to Bahri-Coron’s problem in pierced domains, (preprint).
  3. A. Pistoia, T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear).