Compactness and least energy solutions to the super-Liouville equation on the sphere

In this work, we study the super-Liouville equation on the sphere with positive coefficient functions. We first examine the behavior of the equation under conformal transformations and derive a Pohozaev-type identity, which generalizes the Kazdan-Warner obstruction for the prescribed Gaussian curvature equation. Next, by employing conformal transformations, we obtain an inequality that controls the pointwise norm of the spinor component of the solution in terms of the scalar component. Moreover, we find that the Sobolev energy of the spinor part of the solution is uniformly bounded. Subsequently, we analyze the compactness of the solution space from two perspectives: compactness of solutions in the low-energy regime, and compactness with respect to the Möbius conformal transformation group of the sphere. Finally, by introducing a new natural constraint $\mathcal{A}$, and employing variational methods, we establish the existence of a least-energy solution when the coefficient functions are even. Furthermore, we obtain that the solution is nontrivial, i.e., $\psi \equiv ̸0$, whenever ${\lambda }_1(h_2,h_1)<1$.