Lane Emden problems: asymptotic behavior of low energy nodal solutions

TL;DR

This paper investigates the asymptotic behavior of low energy nodal solutions to the Lane Emden problem in two dimensions, revealing their profiles, estimates, and boundary interactions as the exponent p grows large.

Contribution

It establishes the asymptotic profile and boundary behavior of least energy nodal solutions for the Lane Emden problem as p approaches infinity.

Findings

Least energy nodal solutions satisfy the key asymptotic condition (*)

Solutions' profiles can be characterized as differences of Green's functions

Nodal lines intersect the boundary of the domain for large p

Abstract

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\Omega$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_{\Omega}\abs{\nabla u_p}^2\to 16\pi e\quad\hbox{as}p\rightarrow+\infty\qquad (*)$ and we are interested in the shape and the asymptotic behavior as $p\rightarrow+\infty$. First we prove that (*) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $pu_p$ can be characterized as the difference of two Green's functions and the nodal line intersects the boundary of $\Omega$, for large $p$.