Asymptotic behavior of least energy nodal solutions for biharmonic Lane-Emden problems in dimension four

Abstract

In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem \[ \begin{cases} \Delta^2 u =|u|^{p-1}u \quad &\hbox{in}\;\Omega, \\ u=\frac{\partial u}{\partial \nu}=0 \ \ &\hbox{on}\;\partial\Omega, \end{cases} \] where $\Omega$ is a bounded $C^{4,\alpha}$ domain in $\mathbb{R}^4$ and $p>1$. Among other things, we show that up to a subsequence of $p\to+\infty$, $pu_p(x)\to 64\pi^2\sqrt{e}(G(x,x^+)-G(x,x^-))$, where $x^+\neq x^-\in \Omega$ and $G(x,y)$ is the corresponding Green function of $\Delta^2$. This generalize those results for $-\Delta u=|u|^{p-1}u$ in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincar\'{e}-AN, 30 (2013), 121-140) to the biharmonic case, and also gives an alternative proof of Grossi-Grumiau-Pacella's results without assuming their comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$.