Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Abstract

Let ( M , g ) (M,g) be a smooth compact Riemannian manifold of dimension n ≥ 3 n \ge 3 , and Δ g = − d i v g ∇ \Delta _g = -div_g\nabla be the Laplace-Beltrami operator. Let also 2 ⋆ 2^\star be the critical Sobolev exponent for the embedding of the Sobolev space H 1 2 ( M ) H_1^2(M) into Lebesgue’s spaces, and h h be a smooth function on M M . Elliptic equations of critical Sobolev growth such as ( E ) Δ g u + h u = u 2 ⋆ − 1 \begin{equation*} (E)\qquad \qquad \qquad \qquad \qquad \qquad \Delta _gu + hu = u^{2^\star -1} \qquad \qquad \qquad \qquad \qquad \qquad \end{equation*} have been the target of investigation for decades. A very nice H 1 2 H_1^2 -theory for the asymptotic behaviour of solutions of such equations has been available since the 1980’s. The C 0 C^0 -theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of ( E ) (E) . It was used as a key point by Druet to prove compactness results for equations such as ( E ) (E) . An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of ( E ) (E) . We present such examples in this article.