平面域中Choquard方程最小能量解的Morse指标与谱渐近
Morse index and spectral asymptotics of least-energy solutions to the Choquard equation in planar domains
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中文总结 AI 辅助
研究平面Choquard方程最小能量解的Morse指标与谱渐近，通过变分估计、爆破分析和非局部Pohožaev恒等式，获得前四个特征对的尖锐渐近，并比较光谱指标与Robin函数临界点指标。
中文摘要 AI 辅助
我们研究了平面Choquard方程的最小能量解的Morse指标和谱渐近，该方程为 \begin{equation*} \begin{cases} -\Delta u = \displaystyle\left(\int_{\Omega} \frac{u^{p+1}(y)}{|x - y|^\alpha}dy\right) u^p & \text{in } \Omega, \\\\ u > 0 & \text{in } \Omega, \\\\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{equation*} 其中 $\Omega\subset\mathbb R^2$ 是光滑有界域，$0<\alpha<1$。在某些几何假设下，我们获得了当 $p\to+\infty$ 时线性化问题前四个特征对的尖锐渐近。第一个特征值恰好是 $(2p+1)^{-1}$。第二和第三个特征函数给出了极限气泡的平移模式，其特征值展开由集中点处Robin函数的Hessian矩阵决定。第四个特征函数给出了膨胀模式，其特征值为 $1+3(4-\alpha)/(2p)+o(p^{-1})$。作为推论，我们推导了该解的光谱指标与Robin函数的临界点指标之间的比较。我们的证明结合了变分估计、爆破分析和非局部Pohožaev恒等式。
英文摘要
We investigate the Morse index and spectral asymptotics of least-energy solutions to the planar Choquard equation \begin{equation*} \begin{cases} -Δu = \displaystyle\left(\int_Ω \frac{u^{p+1}(y)}{|x - y|^α}dy\right) u^p & \text{in } Ω, \\ u > 0 & \text{in } Ω, \\ u = 0 & \text{on } \partial Ω, \end{cases} \end{equation*} where $Ω\subset\mathbb R^2$ is a smooth bounded domain and $0<α<1$. Under some geometric assumptions, we obtain sharp asymptotics for the first four eigenpairs of the linearized problem as $p\to+\infty$. The first eigenvalue is exactly $(2p+1)^{-1}$. The second and third eigenfunctions give the translation modes of the limiting bubble, and their eigenvalue expansions are determined by the Hessian of the Robin function at the concentration point. The fourth eigenfunction gives the dilation mode with eigenvalue $1+3(4-α)/(2p)+o(p^{-1})$. As a consequence, we derive a comparison between the spectral indices of the solution and the critical-point indices of the Robin function.The proofs combine variational estimates, blow-up analysis and nonlocal Pohožaev identities.
发表机构
- School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
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