Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in D^1, p (RN)

Abstract 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=D^1, 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: X R Φ: X → R related to the p-Laplacian equation in RN R N in the form aligned u X: - 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 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.