Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

Abstract

In this paper we are interested in the following critical Hartree equation -Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u =\\displaystyle {\\Big (\\int _{\\Omega }\\frac{u^{2_{\\mu }^*} (\\xi )}{|x-\\xi |^{\\mu }}d\\xi \\Big )u^{2_{\\mu }^*-1}}+\\varepsilon u,~~~\ ext {in}~\\Omega ,\\\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ ext {on}~\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Nge 4, 0<mu le 4, varepsilon >0 is a small parameter, Omega is a bounded domain in mathbb {R}^N, and 2_{mu }^*=frac{2N-mu }{N-2} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for varepsilon small.