Abstract
Let(M,g)(M,g)be a smooth compact Riemannian manifold of dimensionnâ„3n \ge 3, andÎg=âdivgâ\Delta _g = -div_g\nablathe Laplace-Beltrami operator. Also let2â2^\starbe the critical Sobolev exponent for the embedding of the Sobolev spaceH12(M)H_1^2(M)into Lebesgue spaces, andhha smooth function onMM. Elliptic equations of critical Sobolev growth like\[Îgu+hu=u2ââ1\Delta _gu + hu = u^{2^\star -1}\]have been the target of investigation for decades. A very niceH12H_1^2-theory for the asymptotic behaviour of solutions of such an equation is available since the 1980âs. In this announcement we present theC0C^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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