On the stability of the critical $p$-Laplace equation

Abstract

For $1<p<n$, it is well-known that non-negative, energy weak solutions to ${\Delta }_{p}u+u^{p^{\ast }-1}=0$ in ${\mathbb{R}}^n$ 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 $u\in {\mathcal{D}}^{1,p}({\mathbb{R}}^n)$ 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.

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Cited by in corpus (1)

• Classification results for bounded positive solutions to the critical $p$-Laplace equation