Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic   operator: existence of extremal functions, breaking positivity and breaking   symmetry
  We investigate Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator on cones, both under Navier and Dirichlet boundary conditions. Moreover, we study existence and qualitative properties of extremal functions. In particular, we show that in some cases extremal functions do change sign; when the domain is the whole space, we prove some breaking symmetry phenomena. 
Paolo Caldiroli and Roberta Musina
math.FA