Local asymptotic nondegeneracy for multi-bubble solutions to the biharmonic Liouville-Gel'fand problem in dimension four

Abstract

We consider the biharmonic Liouville-Gel'fand problem under the Navier boundary condition in four space dimension. Under the nondegeneracy assumption of blow up points of multiple blowing-up solutions, we prove several estimates for the linearized equations and obtain some convergence result. The result can be seen as a weaker version of the asymptotic nondegeneracy of multi-bubble solutions, which was recently established by Grossi-Ohtsuka-Suzuki in two-dimensional Laplacian case.