Symmetry and Concentration Behavior of Ground State in Axially Symmetric Domains

where Ω is a domain in RN . When Ω is a bounded domain in RN being convex in the zi direction and symmetric with respect to the hyperplane {zi = 0}, the famous theorem by Gidas, Ni, and Nirenberg [6] (or see Han and Lin [7]): if u is a positive solution of (1.1) belonging to C2(Ω) ∩C(Ω), then u is axial symmetric in zi. However, the axially symmetry of positive solution generally fails if Ω is not convex in the zi direction. For instance, Dancer [5], Byeon [2, 3], and Jimbo [8] proved that (1.1) in axially symmetric dumbbell-type domain has nonaxially symmetric positive solutions. Wang and Wu [13] and Wu [15] showed the same result in a finite strip with hole. In this paper, we want to show that the symmetry and concentration behavior of ground-state solutions in axially symmetric bounded domains Ω(r) (will be defined later), where the domains Ω(r) are different from those of Dancer [5], Byeon [2, 3], Jimbo [8], and are extensions of Wang and Wu [13] and Wu [15]. The definition of ground-state solution of (1.1) is stated as follows. Consider the energy functionals a, b, and J in H 0 (Ω),