Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball

Abstract

In [41], it was shown that the following singularly perturbed Dirichlet problem e 2 Δu - u + |u| p- u = 0, in Ω, u = 0 on ∂Ω has a nodal solution u e which has the least energy among all nodal solutions. Moreover, it is shown that u e has exactly one local maximum point P e 1 with a positive value and one local minimum point P e 2 with a negative value, and as e → 0, formula math where φ(P 1 , P 2 ) = min(}P 1 - P 2 }/2,d(P 1 , ∂Ω ), d(P 2 , ∂Ω)). The following question naturally arises: where is the nodal surface {u e (x) = 0}? In this paper, we give an answer in the case of the unit ball Ω = B 1 (0). In particular, we show that for e sufficiently small, Pf, P e 2 and the origin must lie on a line. Without loss of generality, we may assume that this line is the x 1 -axis. Then u e must be even in Xj, j = 2,..., N, and odd in x 1 . As a consequence, we show that {u e (x) = 0} = {x ∈ B 1 (0) x 1 = 0}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P e 1 , P e 2 and the origin must lie on the x 1 -axis and u e must be even in Xj, j = 2,..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u e (which implies the odd symmetry of u e in x 1 ). Similar results are also proved for the problem with Neumann boundary conditions.