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 \[ \begin{cases} Δ^2 u =|u|^{p-1}u \quad &\hbox{in}\;Ω, \\ u=\frac{\partial u}{\partial ν}=0 \ \ &\hbox{on}\;\partialΩ, \end{cases} \] 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 $p\to +\infty$, $p{u}_p(x)\to 64{\pi }^2\sqrt{e}(G(x,x^{+})-G(x,x^{-}))$, where $x^{+}\ne x^{-}\in \Omega$ 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.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 $p(\Vert u_p^{+}{\Vert }_{\infty }-\Vert u_p^{-}{\Vert }_{\infty })=O(1)$.