Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in D1,p( RN)

We prove decay estimates for solutions of quasilinear elliptic equations in the whole RN R N of the form aligned u∈ X: - div \,A(x,∇ u)=a(x) f(x,u), aligned u ∈ X : - div A ( x , ∇ u ) = a ( x ) f ( x , u ) , where X=D1,p( RN) X = D 1 , p ( R N ) is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional Φ : X→ R Φ : X → R related to the p-Laplacian equation in RN R N in the form aligned u∈ X: -Δ ₚ u=a(x) g(u), aligned u ∈ X : - Δ p u = a ( x ) g ( u ) , saying that for the subspace V of bounded continuous functions with weight 1+|x|N-p/p, 1 + | x | N - p p , a local minimizer of Φ Φ in the finer V topology is also a local minimizer in the X -topology. Global L^∞ L ∞ -estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.