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

Abstract

Let(M,g)(M,g)be a smooth compact Riemannian manifold of dimensionn≥3n \ge 3, andΔg=−divg∇\Delta _g = -div_g\nablathe Laplace-Beltrami operator. Also let2⋆2^\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\[Δgu+hu=u2⋆−1\Delta _gu + hu = u^{2^\star -1}\]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 1980’s. 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.