Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Abstract

This paper is twofold. In the first part, combining the nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of multi-peak positive solutions to the singularly perturbation problems − 1 and some mild assumptions on the function V . The main difficulties are from the nonlocal operator mixed the nonlocal term, which cause the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single fractional Kirchhoff equation. In the second part, under some assumptions on V , we show the local uniqueness of positive multi-peak solutions by using the local Pohozǎev identity.

Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations

“…Yang [11] proved that for dimensions N > 4s, uniqueness breaks down, i.e., there exist two non-degenerate positive solutions that appear to be completely different from the results of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. They derived the existence of solutions to the singularly perturbed problems [12,13] by combining the Lyapunov-Schmidt reduction approach with this nondegeneracy conclusion. We refer to [14][15][16][17][18][19][20][21][22] for more results of fractional Kirchhoff-type equations employing variational methods.…”