Normalized solutions to the fractional Kirchhoff equations with combined nonlinearities

2021/04/13 by Liu, Lintao, Chen, Haibo, Yang, Jie
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.06053

Abstract

In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation (a+b∫_ℝ³|(-Δ)^(s)/(2)u|²dx)(-Δ)^su=λu+μ|u|^q-2u+|u|^p-2u \hboxin ℝ³, with a prescribed mass ∫_ℝ³|u|²dx=c², where s∈(0, 1), a, b, c>0, 20 and λ∈ℝ as a Lagrange multiplier. Under different assumptions on q0 and μ>0, we prove some existence results about the normalized solutions. Our results extend the results of Luo and Zhang (Calc. Var. Partial Differential Equations 59, 1-35, 2020) to the fractional Kirchhoff equations. Moreover, we give some results about the behavior of the normalized solutions obtained above as μ→0⁺.

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