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.