Existence of minimizers of the stability constant of Caffarelli–Kohn–Nirenberg inequality

Abstract In this paper, we prove the existence of minimizers for the sharp stability constant of Caffarelli–Kohn–Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)$ (which lies above the well-known Felli–Schneider curve $b_{\mathrm{FS}}(a)$ ), extending the work of Wei and Wu [Math. Z., 2024] to a slightly larger region. Moreover, we provide an upper bound for the Caffarelli–Kohn–Nirenberg inequality with an explicit sharp constant, which may have its own interest.