Morse index and spectral asymptotics of least-energy solutions to the Choquard equation in planar domains

Abstract

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.