2025/04/10 by Jin, Zhen-Feng, Wang, Guotao, Zhang, Weimin
#35J35 #35J48 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.07506
In this paper, we consider the existence of normalized solutions for the following biharmonic nonlinear Schrödinger system \begincases Δ2u+α1Δu+λu=βr1|u|^r1-2|v|^r2 u amp; \text in ℝN,
Δ2v+α2Δv+λv=βr2|u|^r1|v|^r2-2 v amp; \text in ℝN,
∫ℝN (u2+v2)\ud x=ρ2, \endcases where Δ2u=Δ(Δu) is the biharmonic operator, α1, α2, β>0, r1, r2>1, N≥ 1. ρ2 stands for the prescribed mass, and λ∈ℝ arises as a Lagrange multiplier. Such single constraint permits mass transformation in two materials. When r1+r2≤ 2+(8)/(N), we obtain a dichotomy result with respect to the mass for the existence of nontrivial ground states. Especially when α1=α2, the ground state exists for all ρ>0 if and only if r1+r2<min\max\4, 2+(8)/(N+1)\, 2+(8)/(N)\. When r1+r2∈(2+(8)/(N), \frac2N(N-4)+) and N≥ 2, we obtain the existence of radial nontrivial mountain pass solution for sufficiently small ρ>0.