Topological degree for singular N-mean field equations
Abstract
We determine the Leray--Schauder degree of the singular Dirichlet N-mean field equation on a smooth bounded domain in ℝN, N≥2, for every positive nonresonant parameter and singular exponents −1<αi≤0. The formula depends only on the Euler characteristic of the domain and the weights (1+αi)N−1. We also identify the homotopy type of every sufficiently negative variational sublevel with the corresponding weighted barycenter space. We prove quantization of interior concentration masses and exclude boundary concentration and residual diffuse mass. The degree computation uses common descent fields for nearby parameters, bounds for their trajectories, and a finite-dimensional comparison with the Euler characteristic of negative sublevels. Annular averaging and dilation provide local contractions, which are extended to establish the sublevel homotopy equivalence. The arguments apply to the nonlinear, possibly degenerate N-Laplacian with singular weights, without requiring a global Palais--Smale condition or isolation and nondegeneracy of solutions.
Related
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BibTeX
@misc{ahmedou2026topological,
title = {{Topological degree for singular $N$-mean field equations}},
author = {Mohameden Ahmedou and Zhongwei Tang and Heming Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20281},
eprint = {2610.20281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20281},
}
RIS
TY - UNPB
TI - Topological degree for singular N-mean field equations
AU - Mohameden Ahmedou
AU - Zhongwei Tang
AU - Heming Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20281
UR - https://arxiv.org/abs/2610.20281
AB - We determine the Leray--Schauder degree of the singular Dirichlet N-mean field equation on a smooth bounded domain in ℝ^N, N≥2, for every positive nonresonant parameter and singular exponents −1<α_i≤0. The formula depends only on the Euler characteristic of the domain and the weights (1+α_i)^N−1. We also identify the homotopy type of every sufficiently negative variational sublevel with the corresponding weighted barycenter space. We prove quantization of interior concentration masses and exclude boundary concentration and residual diffuse mass. The degree computation uses common descent fields for nearby parameters, bounds for their trajectories, and a finite-dimensional comparison with the Euler characteristic of negative sublevels. Annular averaging and dilation provide local contractions, which are extended to establish the sublevel homotopy equivalence. The arguments apply to the nonlinear, possibly degenerate N-Laplacian with singular weights, without requiring a global Palais--Smale condition or isolation and nondegeneracy of solutions.
ER -