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.

We provide regularity, existence and non existence results for the semilinear subelliptic problem with critical growth View the MathML source−ΔGu=ψα|u|2∗(α)−2ud(ξ)α+λu in ΩΩ, u=0u=0 on ∂Ω∂Ω, where ΔGΔG is a sublaplacian on a Carnot group GG, 0<α<20<α<2, 2∗(α)=2(Q−α)/(Q−2)2∗(α)=2(Q−α)/(Q−2), ΩΩ is a bounded domain of GG, dd is the natural gauge associated with the fundamental solution of −ΔG−ΔG on GG and ψ:=|∇Gd|ψ:=|∇Gd|, ∇G∇G being the subelliptic gradient associated to ΔGΔG, λλ is a real parameter.

In this paper, we consider a large class of critical problems with singular potentials on Carnot groups. In particular, we prove sharp regularity results in Marcinkiewicz spaces and the exact rate of decay of the solutions. Our general results apply, for instance, to the equation satisfied by the extremals of some relevant Hardy-Sobolev type inequalities on Stratified groups.