发表机构
Università di Modena e Reggio Emilia（摩德纳与雷焦艾米利亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平面中临界 $p$-Laplace 方程（$1<p<2$）解的唯一连续延拓性质，由此推广 Esteban-Lions 不存在性及 Struwe 的 bubble 分解结果，并用有限能量解刻画紧性丧失。
AI 中文摘要
对于 $1<p<2$，我们证明了平面区域中方程 $\Delta_pu+f(u)=0$ 的解具有弱和强唯一连续延拓性质，其中 $f$ 连续且满足 $|f(s)|\leq C|s|^{p-1}$：在开集上为零的解恒为零，在单点处无穷阶消失的解也恒为零。因此，我们能够推进一些长期未解的不存在性结果的证明，这些结果目前仅对 $p=2$ 成立，例如 Esteban 和 Lions 的一个著名结果，在此我们证明：对于半平面上具有零 Dirichlet 边界条件的临界 $p$-Laplace 方程，不存在非平凡有限能量解。在平面中，唯一连续延拓为将 Struwe 关于与 Brezis-Nirenberg 问题相关的可能变号 Palais-Smale 序列的 bubble 型分解的经典结果推广到 $p$-Laplace 算子提供了缺失的关键要素，该结果在维度 $N\geq3$ 且 $p\neq2$ 时仍属开放问题。我们通过 $\mathbb R^2$ 中 $\Delta_pu+|u|^{p^*-2}u=0$ 的有限能量解刻画了其紧性丧失的特征。
英文摘要
For $1<p<2$ we prove weak and strong unique continuation properties for the solutions to $Δ_pu+f(u)=0$ in a planar domain, with $f$ continuous and such that $|f(s)|\leq C|s|^{p-1}$: a solution vanishing on an open set vanishes identically, and so does a solution vanishing to infinite order at a single point. We can therefore make progress towards a proof of some long-standing nonexistence results available only for $p=2$, such as a celebrated one of Esteban and Lions, establishing here that for the critical $p$-Laplace equation with zero Dirichlet boundary condition on a half-plane, there are no nontrivial finite energy solutions. In the plane, unique continuation thus provides the missing ingredient for a generalisation to the $p$-Laplacian operator of a classical result of Struwe on the bubble-profile decomposition of possibly sign-changing Palais-Smale sequences associated to the Brezis-Nirenberg problem, which remains open in dimension $N\geq3$ for $p\neq2$. We obtain a characterisation of their loss of compactness in terms of the finite energy solutions of $Δ_pu+|u|^{p^*-2}u=0$ in $\mathbb R^2$.
Comments30 pages
