Nondegeneracy of the solutions for elliptic problem with critical exponent

Abstract

This paper deals with the following nonlinear elliptic equation: −Δu=Q(|y′|,y″)uN+2N−2,u>0,inRN,u∈D1,2(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\Delta u=Q(|y'|,y'')u^{\\frac{N+2}{N-2}},\\,\\,u>0,\\,\\,\ ext{in}\\,{ \\mathbb{R}}^{N},\\,\\,u\\in D^{1,2}({\\mathbb{R}}^{N}), $$\\end{document} where (y′,y″)∈R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(y',y'')\\in {\\mathbb{R}}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}, N≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N\\geq 5$\\end{document}, Q(|y′|,y″)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q(|y'|,y'')$\\end{document} is a bounded nonnegative function in R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}. By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).