Local asymptotic nondegeneracy for multi-bubble solutions to the biharmonic Liouville-Gel’fand problem in dimension four. (English) Zbl 1363.35104

Ohtsuka, Hiroshi; Takahashi, Futoshi

Differ. Integral Equ. 28, No. 7-8, 801-822 (2015).

Summary: 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.

MSC:

35J30	Higher-order elliptic equations
35J35	Variational methods for higher-order elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

blowing-up solution; biharmonic operator; Navier boundary conditions