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Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations

Mengni Li, You Li

TL;DR

This work analyzes the Dirichlet problem for a broad class of fully nonlinear, eigenvalue-type equations with a distance-weighted right-hand side in a uniformly convex domain. By adapting the Perron method and comparison principles, it establishes a complete α-based classification for the existence and nonexistence of viscosity solutions, including sharp lower bounds expressed via the distance to the boundary and, at the critical exponent, log-type refinements. The results cover a wide range of operators (e.g., k-Hessian, Monge-Ampère) and provide global barrier-based estimates, linking boundary degeneracy/singularity to solution behavior. The findings advance the understanding of degenerate/singular fully nonlinear elliptic equations and offer explicit, geometry-informed lower bounds that are relevant for geometric PDEs and boundary layer analyses.

Abstract

This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the right-hand side, this type of equations can be degenerate and (or) singular near the boundary of uniformly convex domains. One highlight is that all cases related to the exponent of the distance function are investigated. Moreover, when viscosity solutions exist, we derive a series of global estimates based on the distance function. The key ingredients of this paper include adaptions of the Perron method and comparison principle as well as constructions of suitable classical sub-solutions and super-solutions.

Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations

TL;DR

This work analyzes the Dirichlet problem for a broad class of fully nonlinear, eigenvalue-type equations with a distance-weighted right-hand side in a uniformly convex domain. By adapting the Perron method and comparison principles, it establishes a complete α-based classification for the existence and nonexistence of viscosity solutions, including sharp lower bounds expressed via the distance to the boundary and, at the critical exponent, log-type refinements. The results cover a wide range of operators (e.g., k-Hessian, Monge-Ampère) and provide global barrier-based estimates, linking boundary degeneracy/singularity to solution behavior. The findings advance the understanding of degenerate/singular fully nonlinear elliptic equations and offer explicit, geometry-informed lower bounds that are relevant for geometric PDEs and boundary layer analyses.

Abstract

This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the right-hand side, this type of equations can be degenerate and (or) singular near the boundary of uniformly convex domains. One highlight is that all cases related to the exponent of the distance function are investigated. Moreover, when viscosity solutions exist, we derive a series of global estimates based on the distance function. The key ingredients of this paper include adaptions of the Perron method and comparison principle as well as constructions of suitable classical sub-solutions and super-solutions.

Paper Structure

This paper contains 17 sections, 12 theorems, 291 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions, eigenvalue analysis, and examples
  3. Review of previous results
  4. Main results and outline of this paper
  5. Preliminaries
  6. Classical solutions and viscosity solutions
  7. Construction of approximation functions
  8. Distance function estimates
  9. Eigenvalue estimates
  10. Passing from classical sub-solution to viscosity solution
  11. Proof of Existence Theorem
  12. Case 1: $\alpha\in[0,+\infty)$
  13. Case 2: $\alpha\in(-b,0)$
  14. Case 3: $\alpha=-b$
  15. Case 4: $\alpha\in(-a-2b,-b)$
  16. ...and 2 more sections

Key Result

Theorem 1.2

Under the assumptions $(\mathbf{A_1})$, $(\mathbf{A_2})$-$(\mathrm{i})$, $(\mathbf{A_3})$ and $(\mathbf{A_4})$, we have a list of existence and estimation conclusions based on varying values of $\alpha$:

Figures (2)

  • Figure 1: Estimate of distance function $d(x)$.
  • Figure 2: Relations among the functions $\mathcal{C}_{z_0}(x)$, $u_0(x)$, and $v_0(x)$.

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2: Existence Theorem
  • Theorem 1.3: Nonexistence Theorem
  • Corollary 1.4: Special Case
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Comparison Principle
  • ...and 17 more