Abstract
In this paper, we investigate the constrained minimization problem e(a):=inf{u∈H,∥u∥22=1}Ea(u), where the energy functional Ea(u)=∫R3(u−Δ+m2 u + Vu2) dx − a2∫R3(|x|−1 * u2)u2 dx with m∈R, a>0, is defined on a Sobolev space H. We show that there exists a threshold a*>0 so that e(a) is achieved if 0<a<a* and has no minimizers if a≥a*. We also investigate the asymptotic behavior of non-negative minimizers of e(a) as a→a*.
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