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

Mingyang Han, Ruijun Wu, Chunqin Zhou

Abstract

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemannian surface $M$ with boundary and with its Euler characteristic $χ(M)<0$. The boundary condition couples a Neumann condition for functions and a chirality 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.

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

Abstract

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemannian surface with boundary and with its Euler characteristic . The boundary condition couples a Neumann condition for functions and a chirality 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.

Paper Structure

This paper contains 13 sections, 9 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Moser-Trudinger inequalities
  4. Spinors
  5. Sobolev spaces for spinors
  6. Weighted Dirac operator
  7. The Existence of Non-trivial Solutions
  8. Local behaviors near $(f,0)$
  9. The Nehari Manifold
  10. Existence via minimax theory
  11. $0<\rho<\lambda^{f}_1$
  12. $\rho \in\left(\lambda^{f}_k, \lambda^{f}_{k+1}\right),k\geq 1$
  13. The $(PS)_c$ sequence

Key Result

Theorem 1.1

Let $(M, g)$ be a compact Riemann surface with a fixed spin structure, and with $\partial M \neq \emptyset$ and $\chi \left( M \right)<0$. Suppose that $h(x) \in C^{\infty}(M)$ and $\lambda(x) \in C^{\infty}(\partial M)$ are negative functions, meanwhile $\mathbf{B}^{\pm}$ is a chirality boundary co

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • Definition 3.4
  • Theorem 3.5
  • ...and 5 more