Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

We study the nodal solutions of the Lane–Emden–Dirichlet problem{−Δu=|u|p−1u,in Ω,u=0,on ∂Ω, where Ω   is a smooth bounded domain in R2R2 and p>1p>1. We consider solutions upup satisfyingp∫Ω|∇up|2→16πeas p→+∞ and we are interested in the shape and the asymptotic behavior as p→+∞p→+∞.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 puppup can be characterized as the difference of two Greenʼs functions and the nodal line intersects the boundary of Ω, for large p.