Critical polyharmonic problems with singular nonlinearities

Abstract

Let us consider the Dirichlet problem {(−Δ)mu=|u|pα−2u|x|α+λuinΩDβu|∂Ω=0for|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<Λm,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 to Λm,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=0for0≤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.