具有临界吸收项的双临界 Neumann 问题最小能量解的阈值现象

中文摘要 AI 辅助

我们研究带有临界吸收项的双临界 Neumann 问题最小能量解的存在性与不存在性。考虑定义在光滑有界区域 $\Omega\subset\mathbb{R}^n$（$n\ge 5$）上的正函数 $u$，满足 $\Omega$ 内方程 $-\Delta u + \lambda u = u^{2^* - 1} -\alpha u^{2^\sharp-1}$，以及边界 $\partial\Omega$ 上的 Neumann 边界条件 $\nabla u\cdot \nu = u^{2^\sharp -1}$，其中 $\lambda>0$，$\alpha\ge0$，$2^*=\frac{2n}{n-2}$ 为临界 Sobolev 指数，$2^\sharp=\frac{2(n-1)}{n-2}$ 为临界迹指数。内部与边界临界指数同时存在，加之这一临界吸收项，在竞争集中机制之间产生了新的相互作用。特别地，吸收项与边界非线性具有相同的临界迹指数，但符号相反且作用于 $\Omega$ 内部。因此，在能量展开中，它与边界平均曲率修正以相同的渐近阶竞争，从而产生尖锐的阈值现象。结合多种分析技巧与几何工具，我们证明了阈值 $\alpha_{0}=\alpha_{0}(\lambda,\Omega)\in(0,+\infty)$ 的存在性，使得当 $\alpha<\alpha_{0}$ 时问题存在最小能量解，当 $\alpha>\alpha_{0}$ 时不存在最小能量解。此外，若 $\alpha_0> C(n)\max_{\partial \Omega} H$，其中 $C(n)$ 为仅依赖于 $n$ 的正常数，$H$ 为 $\partial\Omega$ 上的平均曲率，则问题在 $\alpha =\alpha_0$ 时也有最小能量解。

英文摘要

We investigate the existence and nonexistence of least energy solutions for a doubly critical Neumann problem with a critical absorption term. We consider a positive function $u$ defined on a smooth bounded domain $Ω\subset\mathbb{R}^n$ with $n\ge 5$, satisfying $-Δu + λu = u^{2^* - 1} -αu^{2^\sharp-1}$ inside $Ω$, with Neumann boundary condition $\nabla u\cdot ν= u^{2^\sharp -1}$ on $\partialΩ$, where $λ>0$, $α\ge0$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent and $2^\sharp=\frac{2(n-1)}{n-2}$ denotes the critical trace exponent. The simultaneous presence of the interior and boundary critical exponents together with this critical absorption term creates a new interaction between competing concentration mechanisms. In particular, the absorption term has the same critical trace exponent as the boundary nonlinearity, but with the opposite sign and acting in the interior of $Ω$. Consequently, it competes with the boundary mean curvature correction at the same asymptotic order in the energy expansion, leading to a sharp threshold phenomenon. Combining several analytic techniques and geometric tools, we prove the existence of a threshold value $α_{0}=α_{0}(λ,Ω)\in(0,+\infty)$ such that the problem admits a least energy solution if $α<α_{0}$, and no least energy solution if $α>α_{0}$. Moreover, the problem also has a least energy solution at $α=α_0$ provided $α_0> C(n)\max_{\partial Ω} H$, where $C(n)$ is a positive constant depending only on $n$ and $H$ is the mean curvature on $\partialΩ$.

发表机构

• Key Laboratory of Nonlinear Analysis and Applications, School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院非线性分析与应用重点实验室）
• School of Mathematics and Statistics, Xinyang Normal University（信阳师范大学数学与统计学院）

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