Existence of solution for Schrödinger-Newton system with vanishing magnetic potential
Abstract
In this paper, we study the following Schrödinger-Newton system with a vanishing magnetic potential and an electric potential label0.1 { renewcommandarraystretch1.25 ll (−i∇ +A(x))2u+V(x)u=Φu, & x∈ℝ3, −ΔΦ=|u|2, & x∈ℝ3, . where i is the imaginary unit, A:ℝ3→ ℝ3 is a continuous magnetic potential, and V:ℝ3→ ℝ is a continuous electric potential. First, we show that the ground state energy of system with A(x)≡0 and V(x)≡ V∞>0 is isolated. Using this fact together with a global compactness lemma, a barycenter function, and Brouwer degree theory, we prove that under suitable conditions on A and V, system admits either a ground state solution or a high energy solution. Interestingly, whether a ground state solution exists is significantly affected by the magnetic potential A and the electric potential V. Furthermore, we provide sufficient conditions that guarantee the existence of a ground state solution, as well as conditions under which no ground state solution exists and a high energy solution appears instead.
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BibTeX
@misc{shan2026existence,
title = {{Existence of solution for Schrödinger-Newton system with vanishing magnetic potential}},
author = {Liying Shan and Wei Shuai and Ping Yang and Jianghua Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17945},
eprint = {2610.17945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17945},
}
RIS
TY - UNPB
TI - Existence of solution for Schrödinger-Newton system with vanishing magnetic potential
AU - Liying Shan
AU - Wei Shuai
AU - Ping Yang
AU - Jianghua Ye
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17945
UR - https://arxiv.org/abs/2610.17945
AB - In this paper, we study the following Schrödinger-Newton system with a vanishing magnetic potential and an electric potential label0.1 { renewcommandarraystretch1.25 ll (−i∇ +A(x))^2u+V(x)u=Φu, & x∈ℝ^3, −ΔΦ=|u|^2, & x∈ℝ^3, . where i is the imaginary unit, A:ℝ^3→ ℝ^3 is a continuous magnetic potential, and V:ℝ^3→ ℝ is a continuous electric potential. First, we show that the ground state energy of system with A(x)≡0 and V(x)≡ V_∞>0 is isolated. Using this fact together with a global compactness lemma, a barycenter function, and Brouwer degree theory, we prove that under suitable conditions on A and V, system admits either a ground state solution or a high energy solution. Interestingly, whether a ground state solution exists is significantly affected by the magnetic potential A and the electric potential V. Furthermore, we provide sufficient conditions that guarantee the existence of a ground state solution, as well as conditions under which no ground state solution exists and a high energy solution appears instead.
ER -