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| We consider the biharmonic Liouville-Gel'fand problem under the Navier boundary condition in four space dimension:
\begin{align*}
\begin{cases}
\Delta^2 u = \lambda e^u & \quad \mbox{in} \; \Omega, \\
u = \Delta u = 0 & \quad \mbox{on} \; \partial\Omega.
\end{cases}
\end{align*}
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.
This talk is based on a joint work with H. Ohtsuka (Kanazawa University). |
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