PreprintCC BY 4.0arXiv
Multiple solutions for planar Schrodinger-Poisson system with logarithmic nonlinearity
Abstract
This paper is concerned with the following planar Schrodinger-Poisson system: −Δu +V(x)u+λϕu=uln u2, & in R2, Δϕ=u2, & in R2. where λ∈R is a parameter and V∈ C1(R2,R+) is a coercive potential. Due to the presence of the logarithmic nonlinearity, the manifold method developed by Ruiz [R] is not applicable here. We apply a general minimax principle to an auxiliary functional and prove the existence of a nontrivial solution. Then, by minimizing the energy functional over the set of all nontrivial solutions, we succeed in showing that the system admits a ground state solution. If, in addition, V is radially symmetric, we obtain infinitely many non radial sign-changing solutions.
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BibTeX
@misc{long2026multiple,
title = {{Multiple solutions for planar Schrodinger-Poisson system with logarithmic nonlinearity}},
author = {Wei Long and Changchang Yan and Jianghua Ye},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24043},
eprint = {2609.24043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24043},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiple solutions for planar Schrodinger-Poisson system with logarithmic nonlinearity
AU - Wei Long
AU - Changchang Yan
AU - Jianghua Ye
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24043
UR - https://arxiv.org/abs/2609.24043
AB - This paper is concerned with the following planar Schrodinger-Poisson system: −Δu +V(x)u+λϕu=uln u^2, & in R^2, Δϕ=u^2, & in R^2. where λ∈R is a parameter and V∈ C^1(R^2,R^+) is a coercive potential. Due to the presence of the logarithmic nonlinearity, the manifold method developed by Ruiz [R] is not applicable here. We apply a general minimax principle to an auxiliary functional and prove the existence of a nontrivial solution. Then, by minimizing the energy functional over the set of all nontrivial solutions, we succeed in showing that the system admits a ground state solution. If, in addition, V is radially symmetric, we obtain infinitely many non radial sign-changing solutions.
ER -