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

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u =|u|^{p-1}u \quad &{}\hbox {in}\;\Omega , \\ u=\frac{\partial u}{\partial \nu }=0 \ \ {} &{}\hbox {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

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 $p\rightarrow +\infty$, $pu_p(x)\rightarrow 64\pi ^2\sqrt{e}(G(x,x^+)-G(x,x^-))$, 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 et al. (Ann Inst Henri Poincaré Anal Non Lineaire 30:121–140, 2013) to the biharmonic case, and also gives an alternative proof of Grossi–Grumiau–Pacella’s results without assuming their comparable condition $p(\Vert u_p^+\Vert _{\infty }-\Vert u_p^-\Vert _{\infty })=O(1)$.