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Abstract. This paper concerns semilinear elliptic equations whose nonlinear term has the form W(x)f(u) where W changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part of W is contained in a condition which is shown to be necessary for homogeneous /. More general existence questions are also discussed.

L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On nonhomogeneous elliptic equations involving critical Sobolev exponent by G. TARANTELLO (*)

We study the existence of condensate solutions for the Chern-Simons-Higgs model with the choice of a potential field where both the symmetric and asymmetric vacua occur as ground states [see Hong, Kim, and Pac, Phys. Rev. Lett. 64, 2230 (1990) and Jackiw and Weinberg, ibid. 64, 2234 (1990)]. We show that if the Chern-Simons coupling parameter k is above a critical value, no such solutions can exist, while for k≳0 below this critical value there exist at least two condensate solutions carrying the same quantized energy, as well as electric and magnetic charge. This multiplicity result accounts for the two vacua states present in the model. In fact, as k→0+ it is shown that the two solutions found ‘‘bifurcate’’ from the asymmetric and symmetric vacuum states respectively.

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