𝐷1,𝑝(R𝑁) versus 𝐶𝑏 (R𝑁, 1 + |𝑥|𝑁−𝑝/𝑝−1 𝛼) local minimizers

Abstract

Let 𝑋 = 𝐷1,𝑝(R𝑁) be the Beppo-Levi space (homogeneous Sobolev space) with 2 ≤ 𝑝 < 𝑁, and for 𝑝−1 𝑝 < 𝛼 ≤ 1 let 𝑉𝛼 = 𝑋∩𝐶𝑏 ( R𝑁, 1 +|𝑥| 𝑁−𝑝 𝑝−1 𝛼) be the subspace of bounded continuous functions with weight 1 + |𝑥| 𝑁−𝑝 𝑝−1 𝛼 . In this paper we prove a Brezis-Nirenberg type result for the energy functional 𝛷 ∶ 𝑋 → R related to the quasilinear elliptic equation in R𝑁 of the form 𝑢 ∈ 𝑋 ∶ −𝛥𝑝𝑢 = 𝑎(𝑥)𝑔(𝑢) in R𝑁, which states that a local minimizer of 𝛷 in the 𝑉𝛼-topology must be a local minimizer in the ’’bigger’’ 𝑋-topology. Global 𝐿∞-estimates for solutions of general quasilinear elliptic equations of divergence type in R𝑁 on the one hand, and decay estimates for solutions of 𝑝-Laplace equations via nonlinear Wolff potentials as well as comparison theorems for 𝑝-Laplacian type operators on the other hand play an important role in the proofs.