By Graves L. (ed.)

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Further complications arise in the presence of non-compact symmetry groups acting locally; for instance, in the case of conformal invariance or invariance under scaling. 42 Chapter I. The Direct Methods in the Calculus of Variations Existence of Extremal Functions fOT Sobolev Embeddings A typical example is the case of Sobolev's embedding on a (possibly unbounded) domain il c IRn . 4 Sobolev embeddings. For u E C~(il), k ~ 1, p ~ 1, let lIull~k,P = L llDctulP dx , lal=k (J and let Dk,p (il) denote the completion of C~ (il) in this norm.

5 of the appendix. Hence M is weakly closed in Ht,2(fl). 2. 4» In IVul 2 dx ueH~·2(n) In lul 2 dx u"'o inf of the smallest Dirichlet eigenvalue. 2 . From this, coerciveness of E for A > -Al is immediate. Weak lower semi-continuity of E follows from weak lower semi-continuity of the norm in HJ·2(n) and the Rellich-Kondrakov theorem. 2 therefore E attains its infimum at a point y in M. Remark that since E(u) = E(lul) we may assume that y ~ O. 0) with (v, DE(u)} = In Moreover, letting G( u) G : H~·2(n) -+ = (VuVv + AUv) dx .

That is, we have 4. \ = 39 . \ < 1 . \)2/P) 1 + 0(1) . \ E {O, 1}. \ = 0, we obtain that I ~ 100 for large m; a contradiction. 2 m-co ~ cE(um - --+ E(u). 5) u) = c (E(u m ) - E(u)) and Um --+ E, liminf E(u m ) = I , and u minimizes E in M. Hence also E(u m ) I\um E M. n ). The proof is complete. L. Lions [1; p. ]). 3 Concentration-Compactness Lemma I. tm = 1. tm ~ 1- c ) fOT all m. tm) = 0 . 40 Chapter I. p (lA - Ln dJ-L:n1 + 1(1- Ln dJ-L~1) ~ A) - c . Proof. n (rJ Br(x) dJ-L) of a non-negative measure, introduced by P.

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