$p$-Laplace Lane--Emden方程有限Morse指标解的尖锐Liouville阈值与端点刚性

中文摘要 AI 辅助

我们研究$\mathbb{R}^N$中$N>p\ge 2$的方程$-\Delta_p u = |u|^{q-1}u$，其解在紧集外是稳定的，且不假设符号、有界性或对称性。Damascelli、Farina、Sciunzi和Valdinoci解决了$p>2$的次临界范围，并仅对径向解处理了超临界范围$p^*-1<q<q_c(N,p)$，指出在没有径向对称性时他们无法得出结论。障碍在于$p=2$可用的论证依赖于一个单调性公式，而当$p\ne2$时没有单调类比。我们移除了径向假设：当$p^*-1<q<q_c(N,p)$时，每个在紧集外稳定的$C^1$弱解都是平凡的。从而整个范围$p-1<q<q_c(N,p)$（其中$q\ne p^*-1$）被解决，正如Farina定理对$p=2$所做的那样，并且该范围内每个有限Morse指标解都是平凡的。我们转而迭代比较原理，针对一个显式的双参数上解，将稳定性强制的衰减升级为$p$-调和基本解的衰减；此步骤不使用稳定性，仅使用自相似衰减的小性，并单独陈述。上阈值是尖锐的：对每个$q>q_c(N,p)$，我们构造在紧集外稳定的正有界径向解。在Sobolev端点，每个在紧集外稳定的$C^1$解具有有限能量；对$p>2$，这从Farina、Mercuri和Wille的低Morse指标分类中移除了先验的$\mathcal D^{1,p}$假设，并回答了他们对Aubin--Talenti极值Morse指标的问题：它是一。在上端点，$q_c(N,p)$是$\mathbb R^N\setminus\{0\}$上非零齐次弱解在球外稳定的精确阈值，且在阈值处恰好有两个这样的解。

英文摘要

We study $-Δ_p u = |u|^{q-1}u$ in $\mathbb{R}^N$ with $N>p\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^*-1<q<q_c(N,p)$ only for radial solutions, stating that without radial symmetry they were not able to conclude. The obstruction is that the arguments available for $p=2$ rest on a monotonicity formula with no monotone analogue when $p\ne2$. We remove the radial hypothesis: every $C^1$ weak solution stable outside a compact set is trivial when $p^*-1<q<q_c(N,p)$. The whole range $p-1<q<q_c(N,p)$ with $q\ne p^*-1$ is thereby settled, as it is for $p=2$ by Farina's theorem, and every finite Morse index solution in it is trivial. In its place we iterate the comparison principle against an explicit two-parameter supersolution, upgrading the decay forced by stability to that of the $p$-harmonic fundamental solution; this step uses no stability, only smallness of the self-similar decay, and is stated on its own. The upper threshold is sharp: for every $q>q_c(N,p)$ we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every $C^1$ solution stable outside a compact set has finite energy; for $p>2$ this removes the a priori $\mathcal D^{1,p}$ assumption from the low-Morse-index classification of Farina, Mercuri and Willem and answers their question on the Morse index of the Aubin--Talenti extremals: it is one. At the upper endpoint, $q_c(N,p)$ is the exact threshold for a nonzero homogeneous weak solution on $\mathbb R^N\setminus\{0\}$ to be stable outside a ball, and at the threshold there are exactly two.

发表机构

• Faculty of Economic Mathematics, University of Economics and Law（经济数学学院，经济与法律大学）
• Vietnam National University（越南国家大学）

机构由 AI 辅助整理，请以论文原文为准。

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29 pages