Christopher Grumiau

Asymptotics and symmetries of least energy nodal solutions of Lane-Emden problems with slow growth

In this paper, we consider the Lane-Emden problem [GRAPHICS] where is a bounded domain in R-N and p > 2. First, we prove that, for p close to 2, the solution is unique once we. x the projection on the second eigenspace. From this uniqueness property, we deduce partial symmetries of least energy nodal solutions. We also analyze the asymptotic behavior of least energy nodal solutions as p goes to 2. Namely, any accumulation point of sequences of (renormalized) least energy nodal solutions is a second eigenfunction that minimizes a reduced functional on a reduced Nehari manifold. From this asymptotic behavior, we also deduce an example of symmetry breaking. We use numerics to illustrate our results.

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

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