A 𝐶⁰-theory for the blow-up of second order elliptic equations of critical Sobolev growth

Druet, Olivier¹ ; Hebey, Emmanuel² ; Robert, Frédéric³

¹ Département de Mathématiques, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon cedex 07, France
² Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
³ Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

Electronic research announcements of the American Mathematical Society, Tome 09 (2003), pp. 19-25.

Résumé

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta _g = -div_g\nabla$ the Laplace-Beltrami operator. Also let $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue spaces, and $h$ a smooth function on $M$. Elliptic equations of critical Sobolev growth like \[ \Delta _gu + hu = u^{2^\star -1}\] have been the target of investigation for decades. A very nice $H_1^2$-theory for the asymptotic behaviour of solutions of such an equation is available since the 1980’s. In this announcement we present the $C^0$-theory we have recently developed. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of the above equation.

Détail

DOI : 10.1090/S1079-6762-03-00108-2

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