Lane Emden problems: asymptotic behavior of low energy nodal solutions

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$.

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