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 $N \ge 3$, $2^* = \frac{2N}{N - 2}$, $a(x) \in C(\mathbb{R}^N, \mathbb{R})$ is a positive function that is invariant under the map $x \to -\frac{x}{|x|<sup>2}$. 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.

• The paper establishes the existence of positive Kelvin-invariant solutions under minimal symmetry constraints in variable coefficient critical elliptic equations.
• It introduces a refined profile decomposition method that overcomes loss of compactness by recovering energy compactness below the threshold (2/N)S^(N/2).
• The work highlights how Kelvin symmetry paired with variational techniques enables new approaches to critical PDE analysis and motivates further research.

Existence and Profile Decomposition of Kelvin-Invariant Solutions for Critical Elliptic Equations

Problem Setting and Motivation

The paper studies the existence of positive solutions to the critical nonlinear elliptic equation

$$- \Delta u = a(x) |u|^{2^*-2}u \quad \text{in } \mathbb{R}^N,$$

where $N \geq 3$, $2^* = \frac{2N}{N-2}$ is the critical Sobolev exponent, and $a(x)$ is a positive, continuous function on $\mathbb{R}^N$ satisfying the symmetry $a\left(-\frac{x}{|x|^2}\right) = a(x)$. This symmetry corresponds to invariance under the Kelvin transform, a key conformal transformation swapping the origin and infinity in $\mathbb{R}^N$.

Classical results on existence and compactness of solutions to critical elliptic equations rely on invariance under large, often noncompact, symmetry groups—such as the full conformal group. In contrast, this work focuses on the much more restrictive invariance only under the Kelvin transform, and correspondingly, on solutions invariant under this transformation (Kelvin-invariant solutions). The critical challenge here is the loss of compactness due to invariance and scaling, especially under minimal symmetry.

Main Results: Existence Theory with Minimal Symmetry

The principal contribution is the establishment of positive, Kelvin-invariant solutions to the equation above under significantly weaker symmetry assumptions than previously considered. Specifically, the authors prove:

• Existence of Positive Kelvin-Invariant Solutions: Under mild regularity, positivity, boundedness, and Kelvin symmetry of $a(x)$, together with either a flatness condition at the origin/infinity or a suitable energy integral condition, the equation admits a nontrivial, positive, Kelvin-invariant solution.
• Sharp Energy Threshold and Profile Decomposition: The main technical tool is a refinement of the abstract profile decomposition for Palais–Smale sequences, incorporating both the classical translation-dilation symmetries and the Kelvin transform. This yields a new energy compactness threshold: compactness is recovered below the energy level $\frac{2}{N} S^{N/2}$, where $S$ is the Sobolev constant. Importantly, for sublevel solutions, the lack of full conformal symmetry no longer causes loss of compactness.

The precise result is as follows: If $N \geq 3$0 is positive, bounded above by its value at the origin ($N \geq 3$1), and invariant under $N \geq 3$2, and if either

• $N \geq 3$3 as $N \geq 3$4 (i.e., $N \geq 3$5 is flat at 0 and, via symmetry, also at infinity), or
• a certain explicit energy integral involving $N \geq 3$6 is positive,

then the minimization problem associated to the energy functional constrained to Kelvin-invariant functions has a minimizer. By symmetric criticality, this minimizer yields a legitimate solution to the original PDE.

Variational and Analytical Framework

A careful variational setup is provided in the homogeneous Sobolev space $N \geq 3$7, with the Kelvin transform acting as a unitary, self-adjoint involution. The minimization is carried out on the closed subspace of Kelvin-invariant functions. The authors demonstrate that the associated variational functional is itself Kelvin-invariant, and they identify the natural minimization energy as

$N \geq 3$8

with $N \geq 3$9 denoting the set of Kelvin-invariant functions.

The main technical innovation is a new abstract profile decomposition theorem. The authors define a notion of compatibility between a group of symmetries $2^* = \frac{2N}{N-2}$0 (translations and dilations) and an involutive unitary symmetry $2^* = \frac{2N}{N-2}$1, generalizing the notion of 'dislocations' in the concentration-compactness principle. The profile decomposition theorem asserts that any bounded sequence in the Kelvin-invariant subspace admits a decomposition where the 'bubbles' (profiles) appear in symmetric Kelvin pairs, and the only possible lack of compactness arises precisely when the energy crosses the computed threshold.

Energy Estimates and Compactness Restoration

To demonstrate that the critical level $2^* = \frac{2N}{N-2}$2 is below the loss-of-compactness threshold, explicit test function constructions based on Talenti bubbles and their Kelvin images are utilized. The authors prove sharply that $2^* = \frac{2N}{N-2}$3 under the stated assumptions on $2^* = \frac{2N}{N-2}$4 via careful energy expansions. The analysis shows that, even without full conformal invariance, the Kelvin symmetry suffices to prevent dichotomy and concentration phenomena except at or above the precise threshold dictated by the underlying symmetries.

Implications and Theoretical Significance

This work clarifies the minimal symmetry required for compactness recovery and existence of solutions in critical elliptic problems. Showing that a single noncompact symmetry—here, the Kelvin transform—restores compactness below a precise threshold is conceptually significant. It draws a sharp contrast with the classical reliance on more robust symmetry groups, extending the class of variable coefficient problems amenable to variational analysis.

The refined profile decomposition provides a blueprint for future work on critical elliptic equations under other noncompact, yet minimally sufficient, symmetry actions. The compatibility concept for group actions and involutive symmetries likely has applications beyond the elliptic PDE context, including nonlinear harmonic analysis and geometric variational problems.

Future Directions

Potential future developments include:

• Extending the analysis to broader families of variable coefficient problems or domains, possibly with weaker regularity or decay conditions.
• Generalizing the abstract profile decomposition to multiple or more general involutive symmetries.
• Investigating dynamical analogues (e.g., nonlinear wave/Schrödinger equations) where Kelvin-type invariance may control noncompactness in critical regimes.
• Exploring applications in geometric analysis, for example, in prescribing curvature under minimal symmetry constraints.

Conclusion

This paper rigorously establishes the existence of Kelvin-invariant positive solutions for a broad class of critical elliptic equations with variable coefficients under minimal symmetry conditions. Through a novel profile decomposition incorporating the Kelvin transform, the authors recover compactness at a sharp threshold, thereby broadening the scope of variational techniques in critical elliptic theory. The results illuminate fundamental aspects of symmetry and compactness interplay in PDEs, and open pathways for further analysis in critical nonlinear equations with restricted invariance.