Lane Emden problems: asymptotic behavior of low energy nodal solutions

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain$\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$.