Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well

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