Existence of Kelvin-invariant positive solutions for critical elliptic equations with variable coefficients via profile decomposition

In this paper, we consider the following critical nonlinear elliptic equation: ( 𝒫 𝑎 ) { − Δ ⁢ 𝑢 = 𝑎 ⁡ ( 𝑥 ) ⁢ | 𝑢 | 2 ∗ − 2 ⁢ 𝑢 i n ℝ 𝑁 , 𝑢 ∈ 𝒟 1 , 2 ⁡ ( ℝ 𝑁 ) , where 𝑁 ≥ 3 , 2 ∗ = 2 ⁢ 𝑁 𝑁 − 2 , 𝑎 ⁡ ( 𝑥 ) ∈ 𝐶 ⁡ ( ℝ 𝑁 , ℝ ) is a positive function that is invariant under the map 𝑥 → − 𝑥 | 𝑥 | 2 . Under some assumptions on 𝑎 ⁡ ( 𝑥 ) , we show the existence of a positive solution to ( 𝒫 𝑎 ) that is invariant under the Kelvin transform. The symmetry condition imposed here is substantially weaker than the invariance under a noncompact symmetry group that is typically assumed in the literature. The key to the proof is a classification of the Palais–Smale sequences of the associated energy functional. To this end, we establish a new abstract profile decomposition theorem incorporating symmetries such as the Kelvin transform.