By Jeffrey A.

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**Extra resources for Applied partial differential equations. An introduction**

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63) that Ã2=2? Ã2=2? ÂZ Â Z 2? 2 2? h/Kn < 1, and we have that Ã Z Ã2=2? ÂÂ Z 2? 2? juj dvg C juj dvg 1D 1 By assumption, Â Z Ä 1 M 2? M Ã2=2? juj dvg ÂZ 2? C Ã2=2? juj dvg M M R ? since 22? < 1. 64) implies that M juj2 dvg D 1. h/. 1, u solves g u C hu D u2 ? 8, u is of class C 2;Â and, by the maximum principle, u > 0 in M . This ends the proof of the theorem. As a remark, we actually have that u˛ ! 1/ M M R by (ii). 1/. Then u˛ ! u in H 1 as ˛ ! C1. 52) is sharp or not. The condition turns out to be sharp in the following sense.

By the reflexivity of H 1 , and the compactness of the embedding H 1 Lp , there holds that, up to a subsequence, (i) u˛ * u in H 1 , (ii) u˛ ! u in L2 , and u˛ ! u in Lp as ˛ ! C1, for some u 2 H 1 . u˛ / ! u/ in H 1 as ˛ ! C1. 1/ for all v 2 H 1 , we easily obtain that u is a weak solution of g u C ! 39) in M . u/ D cp ). u/ are smooth and everywhere positive. 6. 6. 33) that Z Z Z Z 2 2 2 2 jru˛ j dvg C ! 39) we get that M jru˛ j2 dvg ! M jrup j2 dvg as ˛ ! C1. In particular, u˛ ! up in H 1 as ˛ !

9. 8 are false. S n ; g/ be the unit n-sphere, and for x0 2 S n and ˇ > 1, we define ux0 ;ˇ W S n ! x0 ; /. n 2/ ? 96) u D u2 1 : g u C 4 ? Moreover, they all have the same energy in the sense that kux0 ;ˇ k2L2? 47). In particular, kux0 ;ˇ kH 1 Ä ƒ for all x0 2 S n , all ˇ > 1, and some ƒ > 0. ˇ 1/ Ã n4 2 and thus, kux0 ;ˇ kL1 ! C1 as ˇ ! 1C . 8 is false in this case. 9. M; g/ be a closed Riemannian n-manifold, n 3. u˛ /˛ of solutions of critical equations like g u C h˛ u D u2 ? 1) where 2? h˛ /˛ is a bounded sequence of functions in L1 which converges in L2 , and thus in Lp for all p > 1.

### Applied partial differential equations. An introduction by Jeffrey A.

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