AI 中文总结
本文针对Heisenberg群上扰动CR Yamabe方程，建立Palais-Smale剖面分解与改进的Folland-Stein-Sobolev不等式，证明其多重正解，解决临界嵌入紧性缺失问题。
AI 中文摘要
本文研究Heisenberg群$\boldsymbol{\reals^n}$上涉及次拉普拉斯算子的非齐次临界非线性方程，证明临界问题的多重正解：$\begin{align*} \boldsymbol{\reals^n} u=|u|^{2^\bigstar-2}u+f(\boldsymbol{\reals^n}) \text{ 在 }\boldsymbol{\reals^n}\text{ 中，} u>0,\boldsymbol{u}\text{ 属于 }S^{1,2}(\boldsymbol{\reals^n})\text{，}\boldsymbol{\reals^n}\text{ 是 }\boldsymbol{\reals^n}\text{ 上的次拉普拉斯算子，}2^\bigstar=\frac{2Q}{Q-2},Q=2n+2,n\boldsymbol{\reals^n},S^{1,2}(\boldsymbol{\reals^n})\text{ 是 }\boldsymbol{\reals^n}\text{ 上的齐次Sobolev空间，}f\text{ 是对偶空间}(S^{1,2}(\boldsymbol{\reals^n}))'\text{ 中的非平凡非负泛函，满足适当的小性条件。上述方程是Heisenberg群上CR Yamabe方程的扰动项。主要困难来自临界Folland-Stein嵌入到临界Lebesgue空间的紧性缺失，为此，本文为关联能量泛函建立Palais-Smale剖面分解，该分解通过能量量化确定紧性缺失发生的精确能级，表明每个非紧Palais-Smale序列可分解为有限个弱相互作用气泡的叠加。作为关键分析要素，本文建立了涉及Morrey范数的改进Folland-Stein-Sobolev不等式，该不等式是基础插值不等式，在检测非紧Palais-Smale序列的集中性方面发挥关键作用。
英文摘要
In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group $\mathbb H^n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}_{\mathbb H^n} u=|u|^{2^\star-2}u+f(ξ) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where $ \mathcal{L}_{\mathbb H^n} $ is the sub-Laplacian on $\mathbb H^n$, $2^\star=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\geq 1$, $S^{1,2}(\mathbb H^n)$ is the homogeneous Sobolev space on $\mathbb H^n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{1,2}(\mathbb H^n))'$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.
Comments30 pages. Comments are welcome
