Normalized solutions for Schrödinger system with quadratic and cubic interactions

Abstract

In this paper, we give a complete study on the existence and non-existence of solutions to the following mixed coupled nonlinear Schrödinger system{−Δu+λ1u=βuv+μ1u3+ρv2uinRN,−Δv+λ2v=β2u2+μ2v3+ρu2vinRN, under the normalized mass conditions ∫RNu2dx=b12 and ∫RNv2dx=b22. Here b1,b2>0 are prescribed constants, N≥1, μ1,μ2,ρ>0, β∈R and the frequencies λ1,λ2 are unknown and will appear as Lagrange multipliers. 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.