Existence, Sharp Boundary Asymptotics, and Stochastic Optimal Control for Semilinear Elliptic Equations with Gradient-Dependent Terms and Singular Weights: Theory, Economic Applications, and Numerical Simulations

TL;DR

This paper develops a comprehensive theory for semilinear elliptic equations with gradient-dependent nonlinearities and singular weights, establishing existence, uniqueness, boundary asymptotics, and applications to stochastic control and economic models.

Contribution

It introduces a unified framework for analyzing such equations, deriving sharp boundary asymptotics, proving convexity of solutions, and connecting them to stochastic control problems with practical applications.

Findings

Established existence and uniqueness of large solutions.

Derived sharp boundary blow-up asymptotics and optimal estimates.

Linked solutions to stochastic optimal control problems with state constraints.

Abstract

We develop a unified framework for semilinear elliptic equations with gradient-dependent nonlinearities and singular weights in strictly convex domains. Considering large solutions of \[ -\Delta u + b(x)\,h(|\nabla u|) + a(x)\,u = f(x) \quad \text{in }\Omega\subset\mathbb{R}^N, \] where $h$ is strictly convex with $h(s)\sim s^{q}$ for $q\in(1,2]$ and $a(x),b(x)$ display prescribed singular behavior near $\partial\Omega$, we establish existence and uniqueness via Perron's method and derive sharp boundary blow-up asymptotics. In particular, we obtain optimal liminf--limsup estimates for the rate $\gamma=(\beta-q+2)/(q-1)$, extending recent analytic techniques for gradient-growth problems. We further prove strict convexity of solutions through the microscopic convexity principle, highlighting geometric effects arising from the interaction between the nonlinear gradient term and the singular weights. A verification theorem is provided, showing that the solution coincides with the value function of an infinite-horizon stochastic optimal control problem with state constraints. Applications to Operational Research and Management Science are discussed, including case studies in inventory control, portfolio risk management, and supply chain modeling, as well as connections with boundary layer theory and diffusion processes. Numerical experiments in one and two dimensions confirm the predicted boundary behavior and geometric structure. Python implementations are included in the Appendix.