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: $$-\mathrm{\Delta }u=a(x)\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u\text{in }{\mathbb{R}}^N,u\in {\mathcal{D}}^{1,2}({\mathbb{R}}^N)$$ where $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$, $a(x)\in C({\mathbb{R}}^N,\mathbb{R})$ is a positive function that is invariant under the map $x\to -\frac{x}{\mathrm{\mid }x{\mathrm{\mid }}^2}$. 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.