On the thresholds of the critical singular anisotropic Trudinger–Moser inequality

Abstract In this paper, we investigate the attainable problem for the critical singular anisotropic Trudinger–Moser inequality. It is shown that extremals for the critical singular anisotropic Trudinger–Moser inequality exist only if the parameter 𝑎 lies in an open interval of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>+</m:mo> </m:msup> </m:math> \mathbb{R}^{+} ; outside this interval, there are no extremals. Our result provides a negative answer to a highly non-trivial remaining problem regarding attainability.