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

In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem $$\{\begin{array}{ll}{\mathrm{\Delta }}^{2}u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u& \text{in}\mathrm{\Omega },\\ u=\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ 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 $p\to +\mathrm{\infty }$, $p{u}_p(x)\to 64{\pi }^2\sqrt{e}(G(x,x^{+})-G(x,x^{-}))$, where $x^{+}\mathrm{\ne }{x}^{-}\in \mathrm{\Omega }$ and $G(x,y)$ is the corresponding Green function of ${\mathrm{\Delta }}^2$. This generalize those results for $-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincaré-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 $p(\mathrm{\parallel }{u}_p^{+}{\mathrm{\parallel }}_{\mathrm{\infty }}-\mathrm{\parallel }{u}_p^{-}{\mathrm{\parallel }}_{\mathrm{\infty }})=O(1)$.