PreprintCC BY 4.0arXiv
Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation
Abstract
Let N≥ 3, 2<q<p<2*, and let V∈ L∞(ℝN) be real-valued and ℤN-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator L=−Δ+V. We prove that there exists λ∞>0 such that, for every λ≥λ∞, the competing-power equation −Δu+V(x)u=|u|p−2u−λ|u|q−2u in ℝN has a nontrivial solution uλ∈ H1(ℝN). Moreover, |uλ|H1(ℝN)+|uλ|L∞(ℝN) ≤ Cλ−1/(q−2). We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation Lv=−|v|q−2v.
Related
- Existence and multiplicity of solutions for a class of new nonlocal problem with indefinite potential
- Superregular solitonic solutions of the defocusing nonlocal nonlinear Schrodinger equation and their novel perspectives
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
- Global well-posedness and scattering for the three-dimensional defocusing cubic Schrödinger equation in Hs, s>1/2
- Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five
BibTeX
@misc{bieganowski2026existence,
title = {{Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation}},
author = {Bartosz Bieganowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23446},
eprint = {2609.23446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23446},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation
AU - Bartosz Bieganowski
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23446
UR - https://arxiv.org/abs/2609.23446
AB - Let N≥ 3, 2<q<p<2^*, and let V∈ L^∞(ℝ^N) be real-valued and ℤ^N-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator L=−Δ+V. We prove that there exists λ_∞>0 such that, for every λ≥λ_∞, the competing-power equation −Δu+V(x)u=|u|^p−2u−λ|u|^q−2u in ℝ^N has a nontrivial solution u_λ∈ H^1(ℝ^N). Moreover, |u_λ|_H^1(ℝ^N)+|u_λ|_L^∞(ℝ^N) ≤ Cλ^−1/(q−2). We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation Lv=−|v|^q−2v.
ER -