Variational Problems Arising In Bose-Einstein Condensation

In this thesis, we study L2-constrained minimization problems for a class of energy func-tionals arising in Bose-Einstein condensation(BEC) in R2. We are mainly concerned with the uniqueness, the symmetry property and the blow up behavior of the minimizers.Consider the following energy functional in R2where the parameter α∈R\{0}, H is a suitable functional space, and the potential V(x) satisfies Functional (1) is the well-known Gross-Pitaevskii (GP) energy functional, and it is used to describe the energy of a BEC sysetem. The parameter a>0(a<0) means that the interactions between the atoms are attractive (repulsive). We are interested in the following minimization problem with attractive interactions (a>0): Our main results concerning the minimization problem (2) are as follows:In Chapter3, we first discuss the relationship between the minimizers of problem (2) and the ground states of the corresponding Schrodinger equations (or Euler-Lagrange equations), and prove that:if ua is a minimizer of problem (2), then it is a ground state of the associated Euler-Lagrange equation, and vice versa. Then, by using the implicit function theorem and the tech-niques of operator perturbation, we also prove that (2) has a unique positive minimizer when the parameter a>0is small.In Chapter4, we consider problem (2) under the ring-shaped potential, i.e., V(x)=(\x\-A)2with A>0. In [48], the authors proved the blow-up behavior of the minimizers with a type of polynomial potential. However, their methods depend heavily on that "V(x) has only finite zero (minimum) points", which then cannot be used directly to our case since the ring-shaped potential has infinitely many zero points. By estimating the energy of e(a) twice we get the optimal energy bound, and then by applying the blow-up analysis techniques, we prove that all the nonnegative minimizers of (2) blow up at a point on{x∈R2;|x|=A}. Moreover, using the uniqueness results in Chapter3, we see that the symmetry breaking occurs in the minimizers as the parameter a approaching from0to a*.In Chapter5, we focus on the potential V(x) which equals to zero in a bounded domain. i.e., assuming that there is a bounded domain Ω(?)R2such that Ω={xR2:V(x)=0}. Comparing with the ring-shaped case, since V(x) equals to zero in the whole Ω, it is more difficult to analyze the blow up behavior for the minimizers of problem (2). Based on the argument of Chapter4, we first prove the precise lower bound at infinity of the minimizers by the comparison principle, and then we obtain the optimal energy bound of e(a). Finally, we prove that all the minimizers of (2) blow up at a most centered point of Ω as α↗α*.Chapter6is devoted to the following minimization problem: where q∈(0,2), and Eq(·) is given by Here we assume that V(x) satisfies Let uq(x),q∈(0,2) be a nonnegative minimizer of (3), then for any fixed a∈(0,a*), we prove that limda(q)=da(2) and uq(x) converges to a minimizer of da(2) in H. On the other hand, for any fixed a∈(a*,∞), we first give the optimal energy estimate of da(q), and then prove that uq(x) blows up at a minimum point of V(x) as q↗2. Moreover, if V(x) is the type of polyno-mials, it then can be proved that the blow-up point of uq(x) is the flattest minimum point of V(x).