Normalized solutions for Schrodinger system with quadratic and cubic interactions

摘要

In this paper, we give a complete study on the existence and non-existence of solutions to the following mixed coupled nonlinear Schrodinger system {-delta u +lambda(1)u =beta uv +mu(1)u(3) +rho v(2)u in R-N, -delta v +lambda(2)v =beta/2u(2 )+mu(2)v(3) +rho u(2)v in R-N, under the normalized mass conditions f(RN) u(2)dx = b(1)(2) and f(RN) v2dx = b(2)(2). Here b(1), b(2) > 0 are prescribed constants, N >= 1, mu(1), mu(2,) rho > 0, beta is an element of R and the frequencies lambda(1), lambda(2) are unknown and will appear as Lagrange multipliers. In the one dimension case, the energy functional is bounded from below on the product of L-2 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 L-2-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on b(1) and b(2), we prove the existence of normalized solutions. When N = 3, the energy functional is always unbounded on the product of L-2 spheres. We show that under suitable conditions on b(1) and b(2,) 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 beta -> 0. Finally, we deal with the high dimensional cases N >= 4. Several non-existence results are obtained if beta < 0. When N = 4, beta > 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 beta = 0, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schrodinger system but also leads to a stabilization of the related evolution system.

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