Multiplicity results for the supercritical H´ enon equation ∗ Marino Badiale † and Enrico Serra ‡ July 8, 2004 To Antonio Ambrosetti, for his 60th birthday. Abstract We study the Dirichlet problem for the H´ enon equation −Δu = |x| α u p−1 in the unit ball B ⊂ R N . For N ≥ 4 and α large we prove the existence of positive nonradial solutions for a range of p’s including supercritical values. Mathematics subject classification: 35J20, 35B33. 1 Introduction The equation −Δu = |x| α u p−1 , with α> 0, has been introduced by M. H´ enon [4] as a model to study spherically symmetric clusters of stars. From a mathematical point of view, it is a good * This research was supported by MURST “Variational Methods and Nonlinear Differential Equations” † Dipartimento di Matematica, Universit` a di Torino. Via Carlo Alberto 10, 10123 Torino, Italy. E-mail: badiale@dm.unito.it ‡ Dipartimento di Matematica, Universit`a di Milano, Via Saldini 50. 20133 Milano, Italy. E-mail: enrico.serra@mat.unimi.it 1