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

TL;DR

This paper investigates the qualitative behavior of single blow-up solutions to a nonlocal nonlinear Hartree equation with slightly subcritical exponents, providing estimates on eigenvalues and eigenfunctions, and determining the Morse index of solutions.

Contribution

It offers new estimates on eigenvalues and eigenfunctions for blow-up solutions and analyzes their qualitative properties using local Pohozaev identities and blow-up analysis.

Findings

Derived estimates on the first (n+2)-eigenvalues and eigenfunctions.

Examined the qualitative behavior of eigenpairs for the linearized problem.

Determined the Morse index of single-bubble solutions in a nondegenerate setting.

Abstract

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -\Delta u=(|x|^{-(n-2)}\ast u^{p-\epsilon})u^{p-1-\epsilon}\quad \mbox{in}~~\Omega,~~ u=0\quad \mbox{on}~~\partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $\epsilon>0$ 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 $(\lambda_{i,\epsilon}, v_{i,\epsilon})$ 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.