Strongly Singular Nonlocal Kirchhoff-Type Equations with Variable Exponents: Existence, Regularity, and Renormalized Solutions

TL;DR

This paper develops a comprehensive theory for strongly singular nonlocal Kirchhoff-type equations with variable exponents, establishing existence, regularity, and stability results through novel theorems and computational illustrations.

Contribution

It introduces five original theorems addressing existence, regularity, and solutions of strongly singular nonlocal equations with variable exponents, including renormalized solutions and stability analysis.

Findings

Established existence of solutions via truncation and comparison principles.

Derived optimal regularity and pointwise estimates for solutions.

Proved stability, uniqueness, and Lipschitz dependence of solutions.

Abstract

This work resolves the open problem of strong singularity ($\alpha(z)> 1$) in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with weighted Sobolev spaces and existence via truncation, we develop comparison principles, optimal regularity results, and when classical solutions cease to exist, the construction of renormalized solutions. Building upon these foundations, we establish three advanced results: optimal convergence of truncated sequences to renormalized solutions, refined energy estimates characterizing asymptotic behavior as the truncation parameter vanishes, and a quantitative comparison principle yielding sharp pointwise bounds. Subsequently, we derive sharp two-sided pointwise estimates, a uniqueness theorem with quantitative stability, and Lipschitz continuous dependence of solutions on parameters and boundary data. Each theorem is supported by rigorous proofs employing nonlinear analysis, variational methods, and elliptic regularity theory. A computational illustration visualizes the solution behavior near the boundary and demonstrates convergence of truncated approximations.