Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

Abstract

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem (a+b∫RN|(-Δ)s2u|2dx)(-Δ)su+u=up,inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Big (a+b{\\int _{\\mathbb {R}^{N}}}|(-\\Delta )^{\\frac{s}{2}}u|^2\\mathrm{{d}}x\\Big )(-\\Delta )^su+u=u^p,\\quad \\text {in}\\ \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}where a,b>0, 0<s<1, 1<p<frac{N+2s}{N-2s} and (-Delta )^s is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions N>4s, i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems.