A compactness type result for Paneitz-Branson operators with critical nonlinearity

Abstract

Given $(M,g),$ a compact Riemannian manifold of dimension $n \ge 8,$ we consider positive solutions $u_{\alpha} $ of ${\Delta}^2_gu - div_g(A_{\alpha} du) + a_{\alpha} u = u^{2^\sharp-1}$, where $ A_{\alpha}$ is a smooth, symmetric (2,0) tensor and $a_{\alpha}$ a smooth function. Assuming that $ A_{\alpha}$ and $a_{\alpha}$ converge in a suitable sense as ${\alpha} \rightarrow \infty$, we obtain conditions under which the weak limit of $u_{\alpha}$ is nontrivial.