科研之友
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摘要
In this paper, we look for solutions to the following coupled Schrodinger system: @@@ {-Delta u + lambda(1)u = alpha(1)vertical bar u vertical bar(p-2)u + mu(1)u(3) + rho v(2)u in R-N, @@@ -Delta v + lambda(2)v = alpha(2)vertical bar v vertical bar(p-2)v + mu(2)v(3) + rho u(2)v in R-N, @@@ with the additional conditions integral(N)(R)u(2)dx = b(1)(2) and integral(N)(R)v(2)dx = b(2)(2). Here b(1) and b(2 )> 0 are prescribed, N <= 3, mu(1), mu(2), alpha(1), alpha(2), rho > 0, p is an element of (2, 4) 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, and we prove the existence of normalized ground states under suitable conditions on b(1) and b(2), which are obtained as global minimizers. When N = 3, 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. We also shows the limit behavior of the normalized solutions as alpha(i) -> 0(i = 1, 2). The first solution will disappear and the second solution will converge to the normalized solution of system (1.1) with alpha(i) = 0(i = 1, 2), which has been studied by T. Bartsch, L. Jeanjean, and N. Soave (J. Math. Pures Appl. 2016). Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground states. The results in this paper complement the main results established by X. Luo, X. Yang, and W. Zou (http://arxiv.org/abs/2107.08708), where the authors considered the case N=4.
