Ground states of nonlinear Choquard equations with multi-well potentials

• Published in: Journal of mathematical physics 2016-08, Vol.57 (8), p.1
• Main Authors: Li, Shuai, Xiang, Jianlin, Zeng, Xiaoyu
• Format: Article
• Language: English
• Subjects: Differential equations; Ground state; Nonlinear equations; Physics; Symmetry
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.4961158

In this paper, we study minimizers of the Hartree-type energy functional E a ( u ) ≔ ∫ R N ∇ u ( x ) 2 + V ( x ) u ( x ) 2 d x − a p ∫ R N I α ∗ u ( x ) p u ( x ) p d x , a ≥ 0 under the mass constraint ∫ R N u 2 d x = 1 , where p = N + α + 2 N with α ∈ (0, N) for N ≥ 2 is the mass critical exponent. Here I α denotes the Riesz potential and the trapping potential 0 ≤ V ( x ) ∈ L loc ∞ ( R N ) satisfies lim x → ∞ V ( x ) = ∞ . We prove that minimizers exist if and only if a satisfies a < a ∗ = Q 2 2 ( p − 1 ) , where Q is a positive radially symmetric ground state of − Δ u + u = ( I α ∗ u p ) u p − 2 u in ℝ N . The uniqueness of positive minimizers holds if a > 0 is small enough. The blow-up behavior of positive minimizers as a↗a ∗ is also derived under some general potentials. Especially, we prove that minimizers must blow up at the central point of the biggest inscribed sphere of the set Ω ≔ {x ∈ ℝ N , V(x) = 0} if Ω > 0 .