2021/08/21 by Xiao Luo, Juncheng Wei, Luo, Xiao +5
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2108.09461
In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schrödinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product of L2-spheres, normalized ground states exist and are obtained as global minimizers. When N=2, the energy functional is not always bounded on the product of L2-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on b1 and b2, we prove the existence of normalized solutions. When N=3, the energy functional is always unbounded on the product of L2-spheres. We show that under suitable conditions on b1 and b2, at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as β→ 0. Finally, we deal with the high dimensional cases N≥ 4. Several non-existence results are obtained if β<0. When N=4, β>0, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case β=0, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schrödinger system but also leads to a stabilization of the related evolution system.