Symmetry breaking of extremal functions for the CKN-type inequalities is also studied intensively in the case where p = 2 (see e.

n]) is too small to discuss existences of extremal functions of the best constants.

In the space X we can characterize all extremals in terms of generalized trigonometric functions [7, 9, 10, 11].

In order to characterize the extremals for (4), we use generalized trigonometric functions, which we now briefly define.

Any members of this subset have a legitimate claim to being considered extremal (and therefore interesting).

If the set is not too large (say ten to fifty), it is likely that all members of the set can be proved interesting, using extremal ideas as outlined above, but for very large sets (such as the numbers from one to vigintillion) it is obvious that a sufficient number of properties cannot be assembled.

The extremals which correspond to [Alpha] = 0 are called abnormal, and the extremals which correspond to [Alpha] = 1 are called regular.

In order to show the structure of extremals it is useful to recall the Poisson bracket.

In the study of the existence and nonexistence of extremals of the CKN-type inequalities, as in the study of the Sobolev inequality, the most of difficulty come from the lack of the compactness of the imbedding operators.

Roughly speaking, we shall discuss about the characterizations of the CKN-type inequalities for all [alpha] [member of] R as the variational problems, the existence and nonexistence of the extremal solutions to these variational problems in proper spaces, the exact values and the assymptotic behaviors of the best constants S(p,q,[alpha]) and C(p,q),and so on.

is an

extremal for L, then it is an

extremal for the differential dL also.

We conclude that x(*) is an

extremal for the Lagrangian