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

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.