Precise asymptotic estimates and non-degeneracy of solutions to a biharmonic problem with large exponents in dimension four

We are concerned with the semilinear biharmonic problem under Dirichlet boundary conditions that \begin{equation*} \begin{cases} \Delta^2 u=(u^+)^{p} &{\text{in}~\Omega},\\[0.5mm] u \not\equiv 0 &{\text{in}~\Omega},\\[0.5mm] u=\partial u / \partial \nu = 0 &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^4$ is a smooth bounded domain and $p>1$ is sufficiently large. The basic asymptotic behavior and concentration phenomena of the solutions for this problem have been established in literatures. In this work, we aim to refine some known asymptotic estimates of the solutions to be more explicit, so that we can prove the non-degeneracy of the multi-spikes solutions for general domains. The main methods contain ODE's theory, blow-up analysis, local Pohozaev identities and the use of Green's function and Green's representation.