Rigidity results with applications to best constants and symmetry of   Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities
  We take advantage of a rigidity result for the equation satisfied by an extremal function associated with a special case of the Caffarelli-Kohn-Nirenberg inequalities to get a symmetry result for a larger set of inequali-ties. The main ingredient is a reparametrization of the solutions to the Euler-Lagrange equations and estimates based on the rigidity result. The symmetry results cover a range of parameters which go well beyond the one that can be achieved by symmetrization methods or comparison techniques so far. 
Jean Dolbeault (CEREMADE), Maria J. Esteban (CEREMADE), Stathis   Filippas, Achiles Tertikas
math.AP