On the stability of the critical p-Laplace equation

For $1<p<n$, it is well-known that non-negative, energy weak solutions to Δₚ u + u^p^-1 =0 in Rⁿ are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification is stable up to bubbling. In the present work, we investigate the stability of perturbations of the critical p-Laplace equation for any $1<p<n$, under a condition that prevents bubbling. In particular, we show that any solution u ∈ D1,p(Rⁿ) to such a perturbed equation must be quantitatively close to a bubble. This result generalizes a recent work by the first author, together with Figalli and Maggi (Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6780-6797), in which a sharp quantitative estimate was established for $p=2$. However, our analysis differs completely from theirs and is based on a quantitative P-function approach.