非线性双调和方程的正径向基态：极大值原理与扩张区域逼近

中文摘要 AI 辅助

我们证明了在 $\mathbb R^N$ 中一类非线性双调和方程的非负径向基态的存在性，涵盖了次临界和临界非线性。这些解是通过扩张球上的钳制基态的极限获得的。D. Cassani 和 A. Tarsia 在 [Adv. Nonlinear Anal. 11 (2022)] 中的 Cassani-Tarsia 齐次极大值原理使得逼近态为正。随后，在径向 Nehari 流形上的变分分解排除了所有正半径的零球，因此极限基态在 $\mathbb R^N\setminus\{0\}$ 上为正。当四阶算子分解为两个具有正预解式的二阶算子时，结论升级为在整个 $\mathbb R^N$ 上 $u>0$，包括原点处。我们还给出了一个显式反例，表明将齐次极大值原理推广到具有任意非齐次边界数据的超解的诱人尝试是错误的，即使具有任意大的正零阶系数也是如此。

英文摘要

We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in $\mathbb R^N$, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on $\mathbb R^N\setminus\{0\}$. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to $u>0$ throughout $\mathbb R^N$, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.

发表机构

• Università degli Studi dell’Insubria（因苏布里亚大学）
• RISM-Riemann International School of Mathematics（RISM黎曼国际数学学院）
• China University of Geosciences（中国地质大学）
• Università degli Studi di Udine（乌迪内大学）
• Università di Pisa（比萨大学）

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