Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. VI (2007), 159-183 The equation −u − λ u |x| 2 = |∇ u| p + cf(x): The optimal power BOUMEDIENE ABDELLAOUI AND I RENEO PERAL Abstract. We will consider the following problem −u − λ u |x | 2 = |∇u| p + cf , u > 0 in , where  ⊂ R N is a domain such that 0 ∈ , N ≥ 3, c > 0 and λ> 0. The main objective of this note is to study the precise threshold p + = p + (λ) for which there is no very weak supersolution if p ≥ p + (λ). The optimality of p + (λ) is also proved by showing the solvability of the Dirichlet problem when 1 ≤ p < p + (λ), for c > 0 small enough and f ≥ 0 under some hypotheses that we will prescribe. Mathematics Subject Classification (2000): 35D05 (primary); 35J10, 35J60, 46E30 (secondary). 1. Introduction We consider the linear operator L λ ( · ) ≡−( · ) − λ ( · ) |x | 2 : W 1,2 (R N ) → W −1,2 (R N ), N ≥ 3 and λ> 0. By Hardy inequality L λ is continuous and, moreover, is positive if λ< N = ( N −2 2 ) 2 . We will restrict ourselves to the interval 0 <λ ≤  N where the behavior of L λ is quite peculiar. To have an idea of such a behavior we refer to the papers [11] and [13]. In [11] is proved, among others, the following result. Let  be a bounded domain in R N with 0 ∈ . Consider the problem L λ (u ) = f in , u = 0 on ∂, (P) with f ∈ L m (), 1 < m < 2 N N +2 , and λ<λ m, N ≡ N (m−1)( N −2m) m 2 . Then the weak solution u belongs to W 1,m ∗ 0 (),m ∗ = mN N −m . Moreover the result is optimal. Both authors supported by project MTM2004-02223, M.C.E. Spain. Received June 28, 2006; accepted in revised form February 20, 2007.