Threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrodinger equations

We study ground states of mass critical Schrodinger equations with spatially inhomogeneous nonlinearities in R-2 by analyzing the associated L-2-constraint Gross-Pitaevskii energy functionals. In contrast to the homogeneous case where m(x) equivalent to 1, we prove that both the existence and nonexistence of ground states may occur at the threshold a* depending on the inhomogeneity of m(x). Under some assumptions on m(x) and the external potential V(x), the uniqueness and radial symmetry of ground states are analyzed for almost every a is an element of[0, a*). When there is no ground state at the threshold a*, the limit behavior of ground states as a NE arrow a* is also investigated if V(x) reaches its global minimum in a domain Omega with positive Lebesgue measure and m(x) attains its global maximum at finite points. We show that all the mass concentrates at a flattest global maximum of m(x) within Omega. Published by AIP Publishing.