Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

Abstract

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): $(E_p) \left \{ \begin{array}{ll} \Delta^2 u = u^p & \hbox{in} \; \Omega, \\ u > 0 & \hbox{in} \; \Omega, \\ u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right.$ as p gets large, where Ω is a smooth bounded domain in R ⁴. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.