Minimizers of Hartree Type Functionals and Related Interpolation Inequalities.

In this work we are concerned with the minimization problem associated to the best constant Cn,α in the following interpolation inequality <DFORMULA>∫∫<FRACTION><NUM>|u(x)|α|u(y)|α</NUM><DEN>|x-y|n-2</DEN></FRACTION>dxdy≤Cn,α|∇ψ|L2nα-(n+2)|ψ|L2(n+2)-(n-2)α, u∈H1(Rn)</DFORMULA> for 2≤α<1+4/(n-2) and n≥3. The corresponding variational problem is equivalent to the problem of existence of a ground state solution to the nonlinear non-local elliptic equation <DFORMULA>-Δu+ωu-(|x|-(n-2)*|u|α)|u|α-2u = 0.</DFORMULA> We show that the ground state solution is unique and its mass |u|L22 determines the best constant Cn,α. As a consequence of the above results, we obtain sharp sufficient condition for global existence for the L2-critical Hartree equation. [ABSTRACT FROM AUTHOR]