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: \[ - Δu = a(x) |u|2^*-2u in R^N, u ∈ D1,2(R^N) \] where N ≥ 3, 2^* = 2N/N - 2, a(x) ∈ C(RN, 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 the equation 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.