Normalized solutions of L2-supercritical NLS equations with periodic potentials
Abstract
In this paper, we study normalized solutions to the following L2-supercritical nonlinear Schrödinger equation with a periodic potential −Δu +V (x)u + λu=χΩ(x)f(u) in ℝN, u>0 in ℝN, ∫ℝNabsu2 dx =μ. Here N ≥ 1, μ>0 is prescribed, λ∈ ℝ is a Lagrange multiplier, V∈ C(ℝN) is $1$-periodic in x1,...,xN, f ∈ C1(ℝ) is a nonlinearity having L2-supercritical growth at infinity, Ω⊂ ℝN is either a (nonempty) bounded open set with smooth boundary ∂ Ω or the whole space ℝN, and χΩ is the characteristic function of Ω. In both cases, we prove the existence of normalized solutions of mountain pass type when μ>0 is small and f behaves like a power function |t|p−2t with 2+4/N<p<2* at infinity. Moreover, if Ω is bounded, 1 ≤ N ≤ 4 and f has L2-supercritical growth near the origin, then the existence of normalized solutions is obtained for all μ>0. On the other hand, when Ω=ℝN, we prove the existence of solutions corresponding to local minimizers when μ>0 is small. When Ω is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where Ω= ℝN, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.
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BibTeX
@misc{he2026normalized,
title = {{Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials}},
author = {Zhentao He and Norihisa Ikoma and Chao Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02147},
eprint = {2609.02147},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02147},
}
RIS
TY - UNPB
TI - Normalized solutions of L^2-supercritical NLS equations with periodic potentials
AU - Zhentao He
AU - Norihisa Ikoma
AU - Chao Ji
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.02147
UR - https://arxiv.org/abs/2609.02147
AB - In this paper, we study normalized solutions to the following L^2-supercritical nonlinear Schrödinger equation with a periodic potential −Δu +V (x)u + λu=χ_Ω(x)f(u) in ℝ^N, u>0 in ℝ^N, ∫_ℝ^Nabsu^2 dx =μ. Here N ≥ 1, μ>0 is prescribed, λ∈ ℝ is a Lagrange multiplier, V∈ C(ℝ^N) is $1$-periodic in x_1,...,x_N, f ∈ C^1(ℝ) is a nonlinearity having L^2-supercritical growth at infinity, Ω⊂ ℝ^N is either a (nonempty) bounded open set with smooth boundary ∂ Ω or the whole space ℝ^N, and χ_Ω is the characteristic function of Ω. In both cases, we prove the existence of normalized solutions of mountain pass type when μ>0 is small and f behaves like a power function |t|^p−2t with 2+4/N<p<2^* at infinity. Moreover, if Ω is bounded, 1 ≤ N ≤ 4 and f has L^2-supercritical growth near the origin, then the existence of normalized solutions is obtained for all μ>0. On the other hand, when Ω=ℝ^N, we prove the existence of solutions corresponding to local minimizers when μ>0 is small. When Ω is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where Ω= ℝ^N, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.
ER -