Sharp stability of the weighted Sobolev inequality involving divergent operators on the half space

In this paper, we investigate the stability of a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. We employ variational methods to prove the stability in the functional inequality setting. Moreover, overcoming the absence of an explicit extremal function, we employ the asymptotic behavior of extremal functions to calculate some crucial estimates. By utilizing the finite‐dimensional reduction method, we establish a sharp stability result in the critical points setting.