三维单位球中临界 Neumann 问题的变号解
Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball
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中文总结 AI 辅助
本文证明三维单位球中带零Neumann边界条件的临界方程对任意正参数均存在无穷多个非径向变号解，通过内外粘合与有限维约化构造，并利用约化能量展开定位临界点。
中文摘要 AI 辅助
我们考虑在单位球中具有零 Neumann 边界条件的方程 $\Delta u-\mu u+|u|^4u=0$ 的非径向变号解的存在性，其中 $\mu>0$ 为任意固定参数。我们证明，对每个 $\mu>0$，该问题存在无穷多个互不相同的非径向变号解。更精确地说，对每个足够大的偶数整数 $K$，我们通过将 $\mathbb{R}^3$ 中临界方程的某个变号整体解的 $K$ 个对称排列、适当变换的副本粘合起来构造一个解。该构造将内外粘合方案与有限维约化相结合。约化能量的详细展开使我们能够在允许的参数区域内定位一个内部临界点，从而完成构造。
英文摘要
We consider the existence of nonradial sign-changing solutions to the problem $Δu-μu+|u|^4u=0$ with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter $μ>0$. We prove that, for every $μ>0$, this problem admits infinitely many distinct nonradial sign-changing solutions. More precisely, for every sufficiently large even integer $K$, we construct a solution by gluing $K$ symmetrically arranged, suitably transformed copies of a sign-changing entire solution of the critical equation in $\mathbb{R}^3$. The construction combines an inner--outer gluing scheme with a finite-dimensional reduction. A detailed expansion of the reduced energy allows us to locate an interior critical point in the admissible parameter region and thereby complete the construction.
发表机构
- University of Macau（澳门大学）
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