In this paper we study the following coupled Schrödinger system, which can be seen as a critically coupled perturbed Brezis-Nirenberg problem:Here, ⊂ R 4 is a smooth bounded domain, −λ 1 ( ) < λ 1 , λ 2 < 0, μ 1 , μ 2 > 0 and β = 0, where λ 1 ( ) is the first eigenvalue of − with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (that is, 2N N −2 = 4 when N = 4). We show that this critical system has a positive least energy solution for negative β, positive small β and positive large β. For the case in which λ 1 = λ 2 , we obtain the uniqueness of positive least energy solutions. We also study the limit behavior of the least energy solutions in the repulsive case β → −∞, and phase separation is expected. These seem to be the first results for this Schrödinger system in the critical case.

No abstract

We study the following nonlinear Schrödinger system which is related to Bose-Einstein condensate:

No abstract

We study the existence of homoclinic orbits for first order time-dependent Hamiltonian systems ż=JH z (z, t), where H(z, t) depends periodically on t and H z (z, t) is asymptotically linear in z as |z| Q .. We also consider an asymptotically linear Schrödinger equation in R N .