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

Druet, Olivier; Hebey, Emmanuel; Robert, Frédéric

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

A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth

Authors: Olivier Druet, Emmanuel Hebey and Frédéric Robert
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 19-25
MSC (2000): Primary 35J60; Secondary 58J05
DOI: https://doi.org/10.1090/S1079-6762-03-00108-2
Published electronically: February 3, 2003
MathSciNet review: 1988868
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Abstract: 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.

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