奇异 $p$-Laplace 方程的一个改进的强比较原理及其应用
An improved strong comparison principle for singular $p$-Laplace equations, with applications
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中文总结 AI 辅助
本文改进了奇异 $p$-Laplace 方程的强比较原理，将适用条件从依赖维数的 $\frac{2N+2}{N+2}<p<2$ 扩展为 $\frac32<p<2$，并应用于 Gibbons 猜想及单调性定理。
中文摘要 AI 辅助
我们考虑 $f$ 为正且局部 Lipschitz 连续的方程 $-\Delta_p u=f(u)$ 的正弱解。在奇异情形 $1<p<2$ 中，由 Damascelli 和 Sciunzi [Calc.\\ Var.\\ Partial Differential Equations \textbf{25} (2006), 139--159] 建立的 Harnack 型不等式、线性化算子的强最大值原理、临界集的描述以及强比较原理，二十多年来仅在限制条件 $\frac{2N+2}{N+2}<p<2$ 下可用，而该范围随着 $N\to\infty$ 收缩为空集。我们消除了对维数的依赖，并证明了所有这些结果在 $\frac32<p<2$ 时成立。由于强比较原理在 $p$-Laplace 方程的定性理论中被广泛用作黑箱，这一改进具有传播效应。我们特别表明，在 Esposito、Farina、Montoro 和 Sciunzi [Math.\\ Ann.\\ \textbf{382} (2022), 943--974] 对 $-\Delta_pu=f(u)$ 的 Gibbons 猜想的解决中，以及他们在半空间中变号非线性函数的单调性定理 [Calc.\\ Var.\\ Partial Differential Equations \textbf{61} (2022), art.\\ 154] 中，假设 $\frac{2N+2}{N+2}<p<2$ 可以被 $\frac32<p<2$ 替代；论文正文中还给出了进一步的应用。
英文摘要
We consider positive weak solutions of $-Δ_p u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations \textbf{25} (2006), 139--159] have been available, for more than twenty years, only under the restriction $\frac{2N+2}{N+2}<p<2$, a range that shrinks to the empty set as $N\to\infty$. We remove the dependence on the dimension and prove all of these results for $\frac32<p<2$. Since the strong comparison principle is used as a black box throughout the qualitative theory of $p$-Laplace equations, the improvement propagates. We show in particular that the assumption $\frac{2N+2}{N+2}<p<2$ may be replaced by $\frac32<p<2$ in the resolution of Gibbons' conjecture for $-Δ_pu=f(u)$ by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ \textbf{382} (2022), 943--974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations \textbf{61} (2022), art.\ 154]; further applications are given in the body of the paper.
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（越南国立大学）
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