Topological degree for negative fractional Kazdan--Warner equation on finite graphs

Studies on Kazdan–Warner equations on graphs have grown steadily, yet the fractional case remains insufficiently explored. Using topological degree theory, this work investigates the fractional Kazdan–Warner equation in the negative case on connected finite graphs, focusing on the existence and multiplicity of solutions. This work not only extends the earlier result of S. Liu and Yang (2020) to the fractional setting, but also provides a concise proof for the work of Shan and Y. Liu (2025).

💡 Research Summary

This paper presents a comprehensive study of the negative fractional Kazdan-Warner equation on connected finite graphs, formulated as (-Δ)^s u = (h + λ)e^{2u} - c, where c is a negative constant, s ∈ (0,1), and h is a non-positive function with max h = 0. The primary objective is to establish a complete classification of solution existence and multiplicity depending on the real parameter λ.

The core methodology employs topological degree theory. The authors first prove a crucial a priori bound (Lemma 2), showing that all possible solutions are uniformly bounded independently of λ. This allows the definition of the topological degree for the nonlinear operator F(u) = (-Δ)^s u - (h+λ)e^{2u} + c on a large ball B_R in the function space. Through a carefully constructed homotopy that deforms the original equation to a simpler one, they compute this degree (Lemma 3), finding it to be 1 for λ ≤ 0 and 0 for λ > 0.

This degree calculation directly leads to the main theorem (Theorem 1). For λ ≤ 0, the non-zero degree guarantees existence, and a comparison principle proves the solution is unique. The case λ > 0 is more intricate. The authors define a critical parameter Λ*_s ∈ (0, -min h). They show that for sufficiently small λ > 0, the associated energy functional J_λ admits a local minimum, which is a solution. The parameter Λ*_s is defined as the supremum of λ for which such a local minimum exists. The solution structure is then fully characterized: (i) for 0 < λ < Λ*_s, the equation has at least two distinct solutions—one being the local minimizer, and the second one’s existence is forced by the fact that the topological degree is 0, which cannot be accounted for by a single non-degenerate critical point; (ii) for λ = Λ*_s, there is at least one solution; (iii) for λ > Λ*_s, no solutions exist.

The proof for the multiplicity result (0<λ<Λ*_s) ingeniously combines variational methods (to find the first solution as a local minimizer) with topological degree arguments (to deduce the existence of a second solution). The paper not only extends the earlier result of S. Liu and Yang (2020) from the standard Laplacian (s=1) to the fractional Laplacian (0<s<1) but also provides an alternative, concise proof for the recent work of Shan and Y. Liu (2025) on the same fractional problem, demonstrating the power of topological degree theory in discrete analysis. Numerical experiments complement the theoretical findings by illustrating the behavior of the critical threshold Λ*_s.