Nonlinear Analysis 110 (2014) 77–96 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Critical polyharmonic problems with singular nonlinearities Enrico Jannelli ∗ , Annunziata Loiudice Department of Mathematics, University of Bari, via E. Orabona 4, 70125, Bari, Italy article info Article history: Received 6 June 2014 Accepted 23 July 2014 Communicated by Enzo Mitidieri MSC: 35J60 31B30 35B33 Keywords: Polyharmonic problems Critical exponents Singular nonlinearities abstract Let us consider the Dirichlet problem    (−∆) m u = |u| p α −2 u |x| α + λu in Ω D β u| ∂ Ω = 0 for |β |≤ m − 1 where Ω ⊂ R n is a bounded open set containing the origin, n > 2m,0 <α< 2m and p α = 2(n − α)/(n − 2m). We find that, when n ≥ 4m, this problem has a solution for any 0 <λ< Λ m,1 , where Λ m,1 is the first Dirichlet eigenvalue of (−∆) m in Ω, while, when 2m < n < 4m, the solution exists if λ is sufficiently close to Λ m,1 , and we show that these space dimensions are critical in the sense of Pucci–Serrin and Grunau. Moreover, we find corresponding existence and nonexistence results for the Navier problem, i.e. with boundary conditions ∆ j u| ∂ Ω = 0 for 0 ≤ j ≤ m − 1. To achieve our existence results it is crucial to study the behaviour of the radial positive solutions (whose analytic expression is not known) of the limit problem (−∆) m u = u pα −1 |x| −α in the whole space R n . © 2014 Published by Elsevier Ltd. 1. Introduction In this paper we consider the following two critical growth polyharmonic problems: the first, with Dirichlet boundary conditions, is    (−∆) m u = |u| p α −2 u |x| α + λu in Ω D β u| ∂ Ω = 0 for |β |≤ m − 1; (1.1) the second, with Navier boundary conditions, is    (−∆) m u = |u| p α −2 u |x| α + λu in Ω ∆ j u| ∂ Ω = 0 for 0 ≤ j ≤ m − 1. (1.2) Here m ∈ N, Ω ⊂ R n (n > 2m) is a smooth bounded domain, 0 ∈ Ω,λ ∈ R, 0 <α< 2m, and p α = 2(n−α) n−2m . Moreover, in what follows we shall denote by Λ m,1 the first Dirichlet eigenvalue of (−∆) m in Ω; let us recall that Λ m 1,1 turns out to be the first Navier eigenvalue of (−∆) m in Ω. ∗ Corresponding author. Tel.: +39 805442655. E-mail addresses: enrico.jannelli@uniba.it, enricojannelli@gmail.com (E. Jannelli), annunziata.loiudice@uniba.it (A. Loiudice). http://dx.doi.org/10.1016/j.na.2014.07.017 0362-546X/© 2014 Published by Elsevier Ltd.