Existence of an extremal function of critical Sobolev embedding with an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -homogeneous weight

In [ Calc. Var. Partial Differ. Equ. 60 (2021), no. 1, article no. 16 (27 pages)], we revisited a critical Sobolev-type embedding for weighted Sobolev spaces as introduced in [ J. Differ. Equations 255 (2013), no. 11, pp. 4312–4336] by Cabré and Ros-Oton. In the same paper, we announced to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> - homogeneous weights .