临界分数阶 $p$-Hardy-Sobolev 方程：全局紧性与正解的多重性
Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions
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中文总结 AI 辅助
研究临界分数阶 p-Hardy-Sobolev 方程，建立 Palais-Smale 序列的全局紧性，分析两种集中机制，并在小扰动条件下利用极小极大方法证明两个不同正解的存在性。
中文摘要 AI 辅助
我们研究临界分数阶 $p$-Hardy-Sobolev 方程 \begin{equation}\tag{$\mathcal{P}$}\label{a-main} (-\Delta_p)^s u -\mu\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(\alpha)-2}u}{|x|^{\alpha}}+f \\;\mbox{ in }\\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} 其中 $1<p<\infty$，$0<s<1$，$0\leq\alpha<sp<d$，$\mu>0$，$p^*_s(\alpha):= p(d-\alpha)/(d-sp)$ 是临界 Hardy-Sobolev 指数，$f$ 是 $(\mathcal{D}^{s,p}(\mathbb{R}^d))^*$ 中的非平凡非负泛函。我们首先建立与相应能量泛函相关的 Palais-Smale 序列的全局紧性结果。当 $\alpha>0$ 时，紧性的缺失由 Hardy-Sobolev 极限问题的解的伸缩（dilations）来描述。$\alpha=0$ 的情形具有不同的结构：除了 Hardy 剖面（Hardy profiles）之外，当浓度中心相对于其尺度逃逸出 Hardy 奇点时，可能会出现纯 Sobolev 剖面（pure Sobolev profiles）。我们对这两种集中机制进行直接的“中心-尺度”分析，并获得相应的能量分解和剖面分离。作为应用，在 $f$ 满足一个显式小性假设下，我们首先获得方程 \eqref{a-main} 的一个负能量正解。然后，我们基于隐藏凸性构造一条非线性路径，其能量严格低于第一个鼓泡阈值（bubbling threshold）。结合全局紧性定理的极小极大论证，进而得到方程 \eqref{a-main} 的第二个不同的正解。
英文摘要
We study the critical fractional $p$-Hardy-Sobolev equation \begin{equation}\tag{$\mathcal{P}$}\label{a-main} (-Δ_p)^s u -μ\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(α)-2}u}{|x|^α}+f \;\mbox{ in }\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} where $1<p<\infty$, $0<s<1$, $0\leqα<sp<d$, $μ>0$, $p^*_s(α):= p(d-α)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\mathcal{D}^{s,p}(\mathbb{R}^d))^*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $α>0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $α=0$ has a different structure: in addition to Hardy profiles, pure Sobolev profiles may occur when the centre of concentration escapes from the Hardy singularity relative to its scale. We give a direct centre-scale analysis of these two concentration regimes and obtain the corresponding energy decomposition and profile separation. As an application, under an explicit smallness assumption on $f$, we first obtain a positive solution for \eqref{a-main} with negative energy. We then construct a nonlinear path based on hidden convexity whose energy remains strictly below the first bubbling threshold. A minimax argument, combined with the global compactness theorem, then yields a second distinct positive solution for \eqref{a-main}.
发表机构
- Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）
- Gandhi Institute of Technology and Management (GITAM) University（甘地科技与管理大学）
- Krea University（克雷亚大学）
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