AI 中文总结
本文研究具单调系数的临界p-拉普拉斯方程，完成其正整体解的刘维尔分类，建立Schoen型哈纳克不等式，为经典半线性方法提供拟线性对应及替代证明。
AI 中文摘要
我们研究临界p-拉普拉斯方程$ -\Delta_p u = u^{p^*-1} h(u) $（其中$1 < p < n$）的正弱解，这里$h$是正、有界、连续且非递增的函数。首个主要结果是对由$h$确定的紧致族中每个方程的归一化、有界、正整体解进行完整分类：此类解必与Aubin–Talenti剖面重合；且Aubin–Talenti剖面作为解的存在性，意味着$h$在该剖面取到的所有值构成的区间上为常数。随后，利用该分类结果，我们对定义在$B_{3R}$内的非负弱解建立尺度不变的Schoen型估计：$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $。作为直接推论，我们得到完全无限制的刘维尔定理：每个正整体解必与Aubin–Talenti剖面重合。对纯临界方程，我们还证明对应的刘维尔分类等价于Schoen型哈纳克不等式。本文发展的论证构成了开尔文变换和移动球方法的拟线性对应，这类经典方法仅在半线性框架下适用，且本文还为相应的经典重要结果提供了替代证明。
英文摘要
We investigate positive weak solutions of the critical $p$-Laplace equation $$ -Δ_p u = u^{p^*-1} h(u), \qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate $$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $$ for nonnegative weak solutions defined in $B_{3R}$. As a direct corollary, we obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin--Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.
Comments66 Pages. See Appendix B for a concise outline for the pure-power case with p=2
