Critical polyharmonic problems with singular nonlinearities

Abstract

Let us consider the Dirichlet problem {(-Δ)mu=|u| pα-2u/|x|α+λu in Ω D βu|∂Ω = 0 for |β|≤m-1 where Ω ⊂ Rn is a bounded open set containing the origin, n>2m, 0<α<2m and pα = 2(n-α)/(n-2m). We find that, when n ≥ 4m, this problem has a solution for any 0<λ<Λ m,1 where Λm,1 is the first Dirichlet eigenvalue of (-Δ)m in Ω, while, when 2m<n<4m, the solution exists if λ is sufficiently close toΛm,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 Δju| ∂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 (-Δ) mu = upα-1|x|-α in the whole space Rn