On the stability of the critical p-Laplace equation

For 1 < p < n, it is well known that non-negative, energy weak solutions to Δ_p 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 ∈ D^{1,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., 2018 (21) (2018), pp. 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.