发表机构
Faculty of Mathematics and Computer Science, University of Science; Vietnam National University; Institute of Information Technology and Electrical, Electronics Engineering, University of Transport Ho Chi Minh City; Faculty of Economic Mathematics, University of Economics and Law（大学数学与计算机科学学院; 越南国立大学; 胡志明市交通大学信息技术与电气电子工程学院; 经济法律大学经济数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究Finsler p-Laplacian下负幂和指数非线性方程在紧集外稳定解的不存在性，给出尖锐临界阈值，并修复了早期证明的缺陷。
AI 中文摘要
我们研究方程 $\Delta_p^H u=u^{-q}$, $u>0$ 和 $-\Delta_p^H u=e^u$ 在 $\mathbb{R}^N$ 中的解，其中 $\Delta_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ 是与一致凸范数 $H$ 相关的 Finsler $p$-Laplacian，且 $p\ge2$。我们证明了当 $p\le N<\frac{p(p+3)}{p-1}$ 且 $q$ 高于尖锐临界阈值时，负幂方程不存在在紧集外稳定的解；当 $p<N<\frac{p(p+3)}{p-1}$ 时，指数方程不存在在紧集外稳定的解。因此，在这些范围内，两个方程都不存在有限 Morse 指标的解。在边界情形 $N=p$ 时，我们证明了 $-\Delta_N^H u=e^u$ 在紧集外稳定的解恰好是 Ciraolo 和 Li 分类的有限质量解。一个主要成分与稳定性无关：如果 $N>q\tau$，其中 $\tau=\frac{p}{q+p-1}$，则 $\Delta_p^H u=u^{-q}$ 的正外部解在无穷远处不能保持在奇异解的 $\theta$ 倍之上，对任何 $\theta>1$ 都成立。指数方程也有类似的障碍。这两个障碍都通过与具有冻结右端的显式径向障碍函数进行比较来证明。我们还证明了当 $N>p$ 时以及负幂方程当 $N=p>2$ 时的尖锐性。最后，对于具有 $p$-增长的算子 $\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$，我们建立了稳定性论证所需的积分估计，并完全证明了截断和吸收过程的合理性。这些论证还修复了早期关于相应各向同性和各向异性问题的证明中的缺陷。
英文摘要
We study the equations $Δ_p^H u=u^{-q}$, $u>0$, and $-Δ_p^H u=e^u$ in $\mathbb{R}^N$, where $Δ_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a compact set for the negative-power equation when $p\le N<\frac{p(p+3)}{p-1}$ and $q$ lies above the sharp critical threshold, and for the exponential equation when $p<N<\frac{p(p+3)}{p-1}$. Consequently, neither equation admits solutions of finite Morse index in these ranges. In the borderline case $N=p$, we prove that the solutions of $-Δ_N^H u=e^u$ that are stable outside a compact set are precisely the finite-mass solutions classified by Ciraolo and Li. A main ingredient is independent of stability: if $N>qτ$, where $τ=\frac{p}{q+p-1}$, a positive exterior solution of $Δ_p^H u=u^{-q}$ cannot remain above $θ$ times the singular solution near infinity for any $θ>1$. An analogous obstruction holds for the exponential equation. Both obstructions are proved by comparison with explicit radial barriers having frozen right-hand sides. We also prove sharpness when $N>p$, and for the negative-power equation when $N=p>2$. Finally, for operators $\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$ with $p$-growth, we establish the integral estimates needed for the stability arguments with fully justified truncation and absorption procedures. These arguments also repair gaps in earlier proofs for the corresponding isotropic and anisotropic problems.
Comments21 pages
