Existence of Solutions to a super-Liouville equation with Boundary Conditions

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemann surface M with boundary and with its Euler characteristic χ(M)<0. The boundary condition couples a Neumann condition for functions and a chiral boundary condition for spinors. Due to the generality of the equation, we introduce a weighted Dirac operator based on the solution to a related Liouville equation. Then we construct a Nehari manifold according to the spectral decomposition of the weighted Dirac operator, and use minimax theory on this Nehari manifold to show the existence of the non-trivial solutions.