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By Emmanuel Hebey

The ebook bargains an increased model of lectures given at ETH Zürich within the framework of a Nachdiplomvorlesung. Compactness and balance for nonlinear elliptic equations within the inhomogeneous context of closed Riemannian manifolds are investigated, a box shortly present process nice improvement. the writer describes blow-up phenomena and provides the growth revamped the previous years at the topic, giving an up to date description of the hot principles, options, equipment, and theories within the box. specific realization is dedicated to the nonlinear desk bound Schrödinger equation and to its severe formulation.

Intended to be as self-contained as attainable, the ebook is offered to a wide viewers of readers, together with graduate scholars and researchers.

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2. 2. Up to a subsequence we can assume that u˛ * u1 in H 1 , u˛ ! u1 in L2 , u˛ ! 13), u1 2 H 1 , and v˛ D u˛ u1 . u˛ /˛ has no geometric blow-up ? points, then u˛ ! u1 in L2 and there is nothing to prove. u˛ /˛ . Let 0 Ä Á Ä 1 be a smooth cut-off function around x. ı/ for some 0 < ı 1. 1, the v˛ ’s are bounded in H 1 . Hence, Z Z ? Á2 v˛ / dvg D M M Since v˛ ! 1/ : M By H¨older’s inequality, Z Z ? 2 2? Áv˛ /2 jv˛ j2 M M Ä 2 dvg ÂZ kÁv˛ k2L2? 2? Áv˛ /k2L2 Ä Kn2 2? jv˛ j dvg à 2?? ı/ Assuming that Z ?

U/ "ı for all u 2 H 1 such that kukH 1 D ı, where "ı > 0 is independent of u. T0 u0 / < 0 for T0 1 sufficiently large. 38) 2€ u2 where € is the set of continuous paths from 0 to T0 u0 . Then cp > 0. u˛ /˛ for Ip at level cp . u˛ /˛ is bounded in H 1 . 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 !

We let IO W H 1 ! R be the functional given by Z Z 1 1 ? u/ D ? 2 M 2 M and prove first that the following very standard lemma holds true. 3. u˛ /˛ be a Palais-Smale sequence for I˛ . v˛ /˛ is a Palais-Smale sequence for IO , and Z Z Z ? ? M /. 3. 1. Without loss of generality we can assume that u˛ ! u1 in L2 , and that u˛ ! e. For any ' 2 H 1 , there holds that Z Z Z ? '/ D ZM ZM Z M ? ru˛ r'/dvg C h˛ u˛ 'dvg ju˛ j2 2 u˛ 'dvg M M M Z Z Z ? 15) where ‰˛ D ju˛ j2 ? 2 u˛ ju1 j2 ? 2 u1 jv˛ j2 ? 2 v˛ 3 The Lp and H 1 -theories for blow-up 52 and we recall that u˛ D u1 C v˛ .

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