Abstract:In this paper, we investigate the existence of nontrivial least-energy solutions for the super-Liouville equation with positive coefficient functions on the two-dimensional sphere. Firstly, we derive a global Pohozaev-type identity by analyzing the behavior of solutions under conformal transformations, which generalizes the classical Kazdan-Warner obstruction for the two-dimensional Nirenberg problem. Secondly, by exploiting conformal symmetry, we establish a pointwise estimate that bounds the norm of the spinor component by the scalar component, and show that the $H^1 \times H^{1/2}$ energy of the spinor part remains uniformly bounded. As a byproduct of our analysis, parallel techniques are applied to the Dirac-Einstein equations on the 3-sphere, demonstrating that nontrivial solutions are uniformly bounded away from the trivial solution in the $H^1 \times H^{1/2}$ norm. Moreover, the compactness of the solution space is also analyzed from two perspectives: in the low-energy regime, and modulo the action of the Möbius group. Finally, by introducing a new natural constraint $\mathcal{A}$ and employing variational methods, we obtain a supersymmetric generalization of the Moser-Trudinger-Onofri inequality and establish the existence of least-energy solutions for even coefficient functions. In particular, these solutions are shown to be nontrivial provided that a certain spectral parameter associated with the coefficients satisfies $\lambda_1(h_2, h_1) < 1$. Concurrently, we provide a complete classification of nontrivial least-energy solutions in the case of positive constant coefficients.
From: Mingyang Han [view email]
[v1]
Sat, 20 Sep 2025 14:49:49 UTC (38 KB)
[v2]
Tue, 23 Sep 2025 09:23:35 UTC (38 KB)
[v3]
Thu, 25 Sep 2025 09:38:48 UTC (38 KB)
[v4]
Sun, 3 May 2026 16:43:00 UTC (32 KB)
[v5]
Thu, 18 Jun 2026 11:31:05 UTC (60 KB)
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