发表机构
University of Economics and Law; Vietnam National University（经济与法律大学; 越南国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究互补范围 $2p<\sigma<N$ 内临界拟线性 Hartree 方程的尖锐渐近与对称性，揭示了由 $\sigma(p-2)+2p$ 符号控制的相变，并证明解径向对称且严格递减。
AI 中文摘要
我们建立了临界拟线性 Hartree 方程 \\[ -\Delta_pu=\big(|x|^{-\sigma}*u^{p^*_{\sigma}}\big)u^{p^*_{\sigma}-1} \quad\text{在 }\mathbb R^N \text{ 中}, \qquad p^*_{\sigma}=\frac{p(2N-\sigma)}{2(N-p)}, \\] 在互补范围 $2p<\sigma<N$ 内的尖锐渐近与对称性理论，超越了先前适用于 $\sigma\le2p$ 的拟线性理论 [Math. Ann., 391(2):2653--2708, 2025]。该范围在结构上有所不同：$p^*_{\sigma}<p$，因此通过提取因子 $u^{p-1}$ 得到的有效势在 $u\to0$ 时变得奇异。尽管如此，只要 $p^*_{\sigma}>1$，每个非负有限能量弱解自动局部有界。我们的证明提供了一种适用于该互补范围的拟线性正则性机制，而适用于 $\sigma\le2p$ 的位势论途径在此失效。对于满足 $p^*_{\sigma}>1$ 的正有限能量解，我们还揭示了由 $\sigma(p-2)+2p$ 的符号控制的尖锐转变。若该量为正，则经典的 $p$-调和轮廓得以保持。在阈值处，\\[ u(x)\simeq |x|^{-\gamma_h}(\log|x|)^{1/(p-p^*_{\sigma})}, \qquad \gamma_h=\frac{N-p}{p-1}, \\] 并具有匹配的双侧梯度速率。低于阈值时，\\[ u(x)=A_*|x|^{-\gamma_*}+o(|x|^{-\gamma_*}), \qquad \gamma_*=\frac{2(N-p)(\sigma-p)}{p(\sigma-2p)}<\gamma_h, \\] 且重标度后的梯度一致收敛到相应的径向轮廓。在所有三种情形中，解关于某一点径向对称，并沿径向变量严格递减。
英文摘要
We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation \[ -Δ_pu=\big(|x|^{-σ}*u^{p^*_σ}\big)u^{p^*_σ-1} \quad\text{in }\mathbb R^N, \qquad p^*_σ=\frac{p(2N-σ)}{2(N-p)}, \] in the complementary range $2p<σ<N$, beyond the quasilinear theory previously available for $σ\le2p$ [Math. Ann., 391(2):2653--2708, 2025]. This range is structurally different: $p^*_σ<p$, so the effective potential obtained by factoring out $u^{p-1}$ becomes singular as $u\to0$. Nevertheless, whenever $p^*_σ>1$, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for $σ\le2p$ breaks down. For positive finite-energy solutions with $p^*_σ>1$ we moreover uncover a sharp transition governed by the sign of $σ(p-2)+2p$. If this quantity is positive, the classical $p$-harmonic profile persists. At the threshold, \[ u(x)\simeq |x|^{-γ_h}(\log|x|)^{1/(p-p^*_σ)}, \qquad γ_h=\frac{N-p}{p-1}, \] with the matching two-sided gradient rate. Below the threshold, \[ u(x)=A_*|x|^{-γ_*}+o(|x|^{-γ_*}), \qquad γ_*=\frac{2(N-p)(σ-p)}{p(σ-2p)}<γ_h, \] and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.
Comments40 pages
