Existence of Kelvin-Invariant Positive Solutions for Critical Elliptic Equations with Variable Coefficients via Profile Decomposition

Abstract

In this paper, we consider the following critical nonlinear elliptic equation: \[ - Δu = a(x) |u|^{2^*-2}u \quad \text{in } \mathbb{R}^N, \quad u \in \mathcal{D}^{1,2}(\mathbb{R}^N) \] where $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$, $a(x)\in C({\mathbb{R}}^N,\mathbb{R})$ is a positive function that is invariant under the map $x\to -\frac{x}{|x{|}^2}$. 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.

21 pages