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

2025/11/26 by Cannone, Alessandro, Cingolani, Silvia, Yang, Minbo +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.21372

Abstract

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents -Δu=(|x|^-(n-2)∗ u^p-ε)u^p-1-ε in~~Ω,~~ u=0 on~~∂Ω, where Ω is a smooth bounded domain in ℝⁿ for n=3,4,5, ∗ denotes the standard convolution, ε>0 is a small parameter and p=(n+2)/(n-2) is 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,⋯,n+2. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.

Citations

• Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
• Bubble solution for the critical Hartree equation in pierced domain
• Remainder terms of a nonlocal Sobolev inequality1
• Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations
• Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities
• Local Uniqueness of blow-up solutions for critical Hartree equations in bounded domain
• Uniqueness and nondegeneracy of solutions for a critical nonlocal equation
• Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola
• Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation
• Qualitative properties of multi-bubble solutions for nonlinear elliptic\n equations involving critical exponents
• Asymptotic behavior of solutions for nonlinear elliptic problems with\n the fractional Laplacian
• On the number of peaks of the eigenfunctions of the linearized Gel'fand\n problem
• Morse indices of multiple blow-up solutions to the two-dimensional Gel'fand problem
• Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics
• Solutions of an elliptic system with a nearly critical exponent

Related

• Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
• Local Uniqueness of blow-up solutions for critical Hartree equations in bounded domain
• Qualitative analysis to an eigenvalue problem of the Hartree type Brézis-Nirenberg problem
• Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent
• On well-posedness and blow-up in the generalized Hartree equation