Existence and Local Uniqueness of Normalized Multi-Peak Solutions to a Class of Kirchhoff Type Equations

Abstract

 Westudy the existence and local uniqueness of multi-peak solutions to the following Kirchhoff type equations
 −a+bλ |∇uλ|2 ∆uλ + λ+V(x)uλ =βλup λ, uλ ∈ H1(R3), uλ > 0 in R3, R3 with normalized L2-constraint, that is, u2 λ = 1, R3
 where a > 0, p ∈ (1,5) are constants, λ, bλ, βλ > 0 are parameters, V(x): R3 → R1 is a bounded continuous function. Physicists are very interested in normalized solutions. Compared to finding multi-pick solutions to the equation without normalized L2-constraint one is facing here some new difficulties in getting normalized solutions to the equation. We first prove that for the case of 3 < p < 5, there exist sequences {bλ}λ and {βλ}λ such that for any sufficiently large λ > 0, one can construct multi-peak solutions uλ of some given form to the above equation by using the Lyapunov-Schmidt reduction method under some mild assumptions on the function V (x). In the
 proof of the above existence result, we consider the three cases of p = 11/3,3 < p < 11/3 and 11/3 < p < 5separately, which correspond to the cases of mass critical, subcritical and supercritical in physics respectively. Then, applying the blow-up technique and the local Pohozaev identities we obtain a uniqueness result of multi-peak solutions for the case of 3 < p < 5. The difficulties caused by the nonlocal term and normalized L2-constraint are overcome.