Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem

Abstract

In this paper, we study the following fourth order elliptic problem $(E_p)$: \begin{eqnarray*} (E_p) \left \{ \begin{array}{l} \Delta^2 u = u^p \quad \mbox{in} \Omega, u > 0 \quad \mbox{in} \Omega, u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right. \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and $p >1$ is any positive number. We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$. Combining the arguments of Ren-Wei [8] and Wei [10], we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains, they have one or two ``peaks'' away form the boundary. If it happens that the only one peak point appears, we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$ under the Navier boundary condition.