A P-function approach classifies all solutions to the weighted N-Liouville equation for N=2 and all α>-1, and for N≥3 when -1<α≤0, proving uniqueness of radial solutions in that range.
On the stability of the critical $p$-Laplace equation
4 Pith papers cite this work. Polarity classification is still indexing.
4
Pith papers citing it
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.
fields
math.AP 4representative citing papers
Bounded positive local weak solutions of the critical p-Laplace equation in R^n, for all 1<p<n, are classified as p-bubbles via new optimal integral estimates.
Nearly stationary functions for the Brezis-Nirenberg problem on bounded domains lie within a sharp, dimension-dependent distance of a solution plus bubbles, and the optimal exponents are identified.
The paper establishes an anisotropic Struwe decomposition with bubble interaction estimates, a short classification proof, and quantitative stability for perturbations of the anisotropic critical p-Laplace equation.
citing papers explorer
-
On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$
A P-function approach classifies all solutions to the weighted N-Liouville equation for N=2 and all α>-1, and for N≥3 when -1<α≤0, proving uniqueness of radial solutions in that range.
-
Classification results for bounded positive solutions to the critical $p$-Laplace equation
Bounded positive local weak solutions of the critical p-Laplace equation in R^n, for all 1<p<n, are classified as p-bubbles via new optimal integral estimates.
-
Sharp quantitative stability estimates for the Brezis-Nirenberg problem
Nearly stationary functions for the Brezis-Nirenberg problem on bounded domains lie within a sharp, dimension-dependent distance of a solution plus bubbles, and the optimal exponents are identified.
-
On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results
The paper establishes an anisotropic Struwe decomposition with bubble interaction estimates, a short classification proof, and quantitative stability for perturbations of the anisotropic critical p-Laplace equation.