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Let fl be a bounded domain in Iw" with n 23. We are concerned with the problem of existence of a function u satisfying the nonlinear elliptic equationwhere A is a real constant. The exponent p = ( n + 2 ) / ( n -2) is critical from the viewpoint of Sobolev embedding. Indeed solutions of (0.1) correspond to critical points of the functional where F(x, u ) = f," f (x, t ) dt. Note that p + 1 = 2n/(n -2) is the limiting Sobolev exponent for the embedding HA(fl) c Lp+'(R). Since this embedding is not compact, the functional CP does not satisfy the (PS) condition. Hence there are serious difficulties when trying to find critical points by standard variational methods. In fact, there is a sharp contrast between the case p < ( n + 2 ) / ( n -2) for which the Sobolev embedding is compact, and the case p = (n + 2 ) / ( n -2).Many existence results for problem (0.1) are known when p < (n + 2 ) / ( n -2) (see the review article by P. L. Lions [20] and the abundant list of references in [20]). On the other hand, a well-known nonexistence result of Pohozaev [24]

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Bethuel, Fabrice, 1%3-Ginzburg-Landau vortices I Fabrice Bethuel, Halm Brezis, Frederic Helein.p. cm. --(Progress in nonIinear differential equations and their applications ; v. 13) Included bibliographical references and index.

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