Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well. (4th August 2022)

• Record Type: Journal Article
• Authors: Tang, Zhongwei; Zhang, Chengxiang; Zhang, Luyu; Zhou, Luyan
• Is Part Of: Nonlinearity. Volume 35:Number 8(2022)
• Page Start: 4624
• Page End: 4658
• Publication Date: 2022-08-04
• Subjects: nonlinear Schrödinger equation -- normalized multibump solution -- steep potential well; 35J50 -- 35J15 -- 35J60; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac7b61
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We are concerned with the existence of multibump solutions to the nonlinear Schrödinger equation − Δ u + λ a ( x ) u + μ u = | u | 2 σ u in R N with an L 2 -constraint ‖ u ‖ L 2 ( R N ) 2 = ρ in the L 2 -subcritical case σ ∈ (0, 2/ N ) and the L 2 -supercritical case σ ∈ (2/ N, 2*/ N ), where the usual critical Sobolev exponent is 2* = +∞ if N = 1, 2 and 2* = 2 N /( N − 2) if N ⩾ 3. Here μ ∈ R will arise as a Lagrange multiplier, and 0 ⩽ a ∈ L loc ∞ ( R N ) has a bottom int a −1 (0) composed of ℓ 0 ( ℓ 0 ⩾ 1) connected components { Ω i } i = 1 ℓ 0, where int a −1 (0) is the interior of the zero set a − 1 ( 0 ) = { x ∈ R N | a ( x ) = 0 } of a . When ρ is fixed either large in the L 2 -subcritical case or small in the L 2 -supercritical case, we construct a ℓ -bump (1 ⩽ ℓ ⩽ ℓ 0 ) positive normalized solution which is localised at ℓ prescribed components { Ω i } i = 1 ℓ for large λ . The asymptotic profile of the solution is also analysed through taking the limit as λ → +∞, and subsequently as ρ → +∞ in the L 2 -subcritical case or ρ → 0 + in the L 2 -supercritical case. In particular, we find ℓ -bump normalized solutions to the related Dirichlet problem − Δ v + μ v = | v | 2 σ v, v ∈ H 0 1 ( ∪ i = 1 ℓ Ω i ), ∑ i = 1 ℓ ∫ Ω i v 2 = ρ, v | Ω i > 0 for i = 1, …, ℓ .