Critical growth problems with singular nonlinearities on Carnot groups

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

Existence results for critical problems involving p-sub-Laplacians on Carnot groups

We provide existence results for the quasilinear subelliptic Dirichlet problem -\Delta_{p, \mathbb{G}}u = |u|^{p^*-2}u + g(\xi, u) \quad \mbox{in}\, \Omega,\quad u\in S_0^{1,p}(\Omega), where $\Delta_{p,\mathbb{G}}$ is the $p$-sub-Laplacian on a Carnot group $\mathbb{G}$, $p^*= pQ/(Q-p)$ is the critical Sobolev exponent in this context, $\Omega$ is a bounded domain of $\mathbb{G}$ and $g(\xi, u)$ is a subcritical perturbation. By means of standard variational methods adapted to the stratified context, we prove the existence of solutions both in the mountain pass and in the linking case. A crucial ingredient in this abstract framework is the knowledge of the exact rate of decay of the $p$-Sobolev extremals on Carnot groups

Local Behavior of Solutions to Subelliptic Problems with Hardy Potential on Carnot Groups

We determine the exact behavior at the singularity of solutions to semilinear subelliptic problems of the type -ΔGu-μψ2/d2u=f(ξ,u) in Ω , u= 0 on ∂Ω , where ΔG is a sub-Laplacian on a Carnot group G of homogeneous dimension Q, Ω is an open subset of G, 0 ∈ Ω , d is the gauge norm on G, ψ: = |∇Gd| , where ∇Gis the horizontal gradient associated with ΔG, f has at most critical growth and 0 ≤ μ< μ¯ , where μ¯=(Q-2/2)^2 is the best Hardy constant on G

Existence of infinitely many solutions for a class of semilinear subelliptic equations on rational Carnot groups

We establish the existence of infinitely many solutions for the equation -Δ_Gu = f(ξ,u), ξ in G where Δ_G is a sublaplacian on a rational Carnot group G. The function f is assumed to be periodic with respect to a discrete co-compact subgroup of G and satisfy subcritical growth conditions

Critical problems with Hardy potential on Stratified Lie groups

We prove existence and nonexistence results for positive solutions to the subelliptic Brezis-Nirenberg type problem with Hardy potential −ΔGu−μψ2/d2u=u2∗−1+λuinΩ,u=0on∂Ω, extending to the Stratified setting well-known Euclidean results by Jannelli [J. Diff. Equ. 156, 1999]. Here, ΔG is a sub-Laplacian on an arbitrary Carnot group G, Ω is a bounded domain of G, 0∈Ω, d is the ΔG-gauge, ψ:=|ΔGd|, where ΔG is the horizontal gradient associated to ΔG, 0≤μ&lt; ̄μ, where ̄μ=(Q−22)2 is the best Hardy constant on G and λ∈R. The main difficulty in this abstract framework is the lack of knowledge of the ground state solutions to the limit problem −ΔGu−μψ2/d2u=u2∗−1onG, whose explicit expression is not known, except for the case when μ=0 and G is a group of Iwasawa-type. So, a preliminary refined analysis of qualitative properties of solutions to the above problem on the whole space is required, which has independent interest. In particular, Lorentz regularity and a priori decay estimates are obtained

Optimal decay of p-Sobolev extremals on Carnot groups

We determine the sharp asymptotic behavior at infinity of solutions to quasilinear critical problems involving the p-sublaplacian operator Δp,G on a Carnot group G,

Sobolev inequalities with remaider terms for Sublaplacians and other subelliptic operators

We prove that on bounded domains Ω, the usual Sobolev inequality for sublaplacians on Carnot groups can be improved by adding a remainder term, in striking analogy with the euclidean case. We also show analogous results for subelliptic operators like script L sign = Δ x + |x| ^{2α} Δy, α &gt; 0

Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators

We provide the asymptotic behavior of solutions, at the singularity and at infinity, for a class of subelliptic Dirichlet problems with Hardy perturbation and critical nonlinearity of the type −Lαu−μψ^2/d^2u=K(z)|u|^(2∗−2)u in Ω, where Lα=Δx+|x|2αΔy, α&gt;0 is the so-called Grushin operator, Ω is an open subset of R^N, 0∈Ω, d is the gauge norm naturally associated with Lα, ψ:=|∇αd|, where ∇α is the Grushin gradient, K∈L∞ and 0≤μ&lt; mus$, where mus is the best Hardy constant for Lα. Furthermore, we establish some Pohozaev-type non-existence results