Lane Emden problems: asymptotic behavior of low energy nodal solutions

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\mathrm{\Omega }$ in ${\text{IR}}^2$ 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 $p\to +\mathrm{\infty }$. 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 $p{u}_p$ can be characterized as the difference of two Green’s functions and the nodal line intersects the boundary of $\mathrm{\Omega }$, for large $p$.