Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well

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* = 2N/(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 0\quad \text{for}\ i=1,\dots ,\ell .\hfill \end{aligned}\right.\end{equation*}?> − 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 .