Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem 2∆u− u+ |u|p−1u = 0, in Ω, u = 0 on ∂Ω has a nodal solution u which has the least energy among all nodal solutions. Moreover, it is shown that u has exactly one local maximum point P 1 with a positive value and one local minimum point P 2 with a negative value and, as → 0, φ(P 1 , P 2 ) → max (P1,P2)∈Ω×Ω φ(P1, P2), where φ(P1, P2) = min( |P1−P2 2 , d(P1, ∂Ω), d(P2, ∂Ω)). The following question naturally arises: where is the nodal surface {u (x) = 0}? In this paper, we give an answer in the case of the unit ball Ω = B1(0). In particular, we show that for sufficiently small, P 1 , P 2 and the origin must lie on a line. Without loss of generality, we may assume that this line is the x1axis. Then u must be even in xj , j = 2, ..., N , and odd in x1. As a consequence, we show that {u (x) = 0} = {x ∈ B1(0)|x1 = 0}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P 1 , P 2 and the origin must lie on the x1-axis and u must be even in xj , j = 2, ..., N . Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u (which implies the odd symmetry of u in x1). Similar results are also proved for the problem with Neumann boundary conditions.