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

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

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.