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Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

Alessandro Cannone, Silvia Cingolani, Minbo Yang, Shunneng Zhao

TL;DR

The paper analyzes the qualitative structure of single blow-up solutions to the nonlocal Hartree equation with slightly subcritical exponent on bounded domains in dimensions 3–5. By combining local Pohozaev identities with blow-up analysis, it derives precise asymptotics for the first (n+2) eigenpairs of the linearized operator and connects these to the Robin function and Green's function, revealing how curvature of φ governs Morse index through the Hessian eigenvalues. The results show the first eigenvalue converges to a subcritical limit, the next n+1 eigenvalues approach 1 with Hessian-determined corrections, and the (n+2)-th eigenvalue exhibits a distinct asymptotic tied to a two-region nodal structure; together, they yield explicit Morse-index relations and a nondegenerate blow-up behavior. This work extends local results to a nonlocal Hartree setting, highlighting the interplay between nonlocal nonlinearities, blow-up profiles, and spectral stability via Robin-function geometry, with potential implications for stability analysis of singular solutions in nonlocal elliptic problems.

Abstract

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -Δu=(|x|^{-(n-2)}\ast u^{p-ε})u^{p-1-ε}\quad \mbox{in}~~Ω,~~ u=0\quad \mbox{on}~~\partialΩ, \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(λ_{i,ε}, v_{i,ε})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.

Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

TL;DR

The paper analyzes the qualitative structure of single blow-up solutions to the nonlocal Hartree equation with slightly subcritical exponent on bounded domains in dimensions 3–5. By combining local Pohozaev identities with blow-up analysis, it derives precise asymptotics for the first (n+2) eigenpairs of the linearized operator and connects these to the Robin function and Green's function, revealing how curvature of φ governs Morse index through the Hessian eigenvalues. The results show the first eigenvalue converges to a subcritical limit, the next n+1 eigenvalues approach 1 with Hessian-determined corrections, and the (n+2)-th eigenvalue exhibits a distinct asymptotic tied to a two-region nodal structure; together, they yield explicit Morse-index relations and a nondegenerate blow-up behavior. This work extends local results to a nonlocal Hartree setting, highlighting the interplay between nonlocal nonlinearities, blow-up profiles, and spectral stability via Robin-function geometry, with potential implications for stability analysis of singular solutions in nonlocal elliptic problems.

Abstract

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -Δu=(|x|^{-(n-2)}\ast u^{p-ε})u^{p-1-ε}\quad \mbox{in}~~Ω,~~ u=0\quad \mbox{on}~~\partialΩ, \end{equation*} where is a smooth bounded domain in for , denotes the standard convolution, is a small parameter and is energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first -eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs to the linearied problem of the above nonlocal equations for . As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.

Paper Structure

This paper contains 9 sections, 24 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Structure of the paper
  4. Decay estimate of the rescaled eigenfunction $\tilde{v}_{i,\varepsilon}$
  5. Asymptotic behavior of the eigenpair $(\lambda_{1,\varepsilon},v_{1,\varepsilon})$
  6. Estimates for the eigenvalues $\lambda_{i,\varepsilon}$ for $i=2,\dots,n+1$
  7. Asyptotic behaviour of the eigenfunctions $v_{i,\varepsilon}$ for $i=2,\cdots,n+1$
  8. Asymptotic behaviour of the eigenvalues $\lambda_{i,\varepsilon}$ for $i=2,\cdots,n+1$
  9. Qualitative characteristics for $(n+2)$-th eigenvalue and corresponding to eigenfunction

Key Result

Theorem 1.1

Assume that $n=3,4,5$, $p=\frac{n+2}{n-2}$ and $\varepsilon$ is sufficiently small. Then Moreover, if $\varepsilon$ is small enough, the eigenvalue $\lambda_{1,\varepsilon}$ is simple and with Here $C_{n,n-2}$ is defined in defhlsbc and the convergence is in $C^{1,\tilde{\alpha}}(\omega)$ with $\omega$ any subdomain of $\Omega$ not containing $x_0$, and $G$ denotes the Green's function of the L

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 38 more