Lane Emden problems: asymptotic behavior of low energy nodal solutions

Abstract

We study the nodal solutions of the Lane Emden Dirichlet problem $-Δu = |u|^{p-1}u with DBC on a smooth bounded domain $Ω$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_Ω\abs{\nabla u_p}^2\to 16πe\quad\hbox{as}p\rightarrow+\infty\qquad (*)$ 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 $Ω$, for large $p$.

Annales de l'Institut Henri Poincaré, 2012