On the existence of extremal functions in Sobolev embedding theorems with critical exponents

Abstract

Abstract. Sufficient conditions for the existence of extremal functions in Sobolevtype inequalities on manifolds with or without boundary are established. Some of these conditions are shown to be sharp. Similar results for embeddings in some weighted L q -spaces are obtained.

The extremal problem for Sobolev inequalities with upper order remainder terms

“…Considerable work has been devoted to the study of extremal functions to sharp Sobolev inequalities in recent decades (see Aubin [2], Aubin and Li [4], Brouttelande [9], Carleson and Chang [11], Collion, Hebey and Vaugon [13], Demyanov and Nazarov [14], Djadli and Druet [15], Druet, Hebey and Vaugon [18], Hebey [21,23], Humbert [26], Li [27], Struwe [33] and Zhu [36]). Such functions are connected, for instance, with the computation of ground state energy in some physical models.…”