Existence of Kelvin-invariant positive solutions for critical elliptic equations with variable coefficients via profile decomposition
Year: 2026
Type: article
Abstract: In this paper, we consider the following critical nonlinear elliptic equation: ( π« π ) { β Ξ β’ π’ = π β‘ ( π₯ ) β’ | π’ | 2 β β 2 β’ π’ i n β π , π’ β π 1 , 2 β‘ ( β π ) , where π β₯ 3 , 2 β = 2 β’ π π β 2 , π β‘ ( π₯ ) β πΆ β‘ ( β π , β ) is a positive function that is invariant under the map π₯ β β π₯ | π₯ | 2 . Under some assumptions on π β‘ ( π₯ ) , we show the existence of a positive solution to ( π« π ) 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 PalaisβSmale sequences of the associated energy functional. To this end, we establish a new abstract profile decomposition theorem incorporating symmetries such as the Kelvin transform.
Authors: Yohei Sato, Reo Suzuki
Institution: Saitama University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7167671891