Existence, Sharp Boundary Asymptotics, and Stochastic Optimal Control for Semilinear Elliptic Equations with Gradient-Dependent Terms and Singular Weights

Abstract

We establish existence, uniqueness, and precise boundary asymptotic behavior for large solutions to the semilinear elliptic equation \begin{equation*} -Δu+b(x)h(|\nabla u|)+a(x)u=f(x) \end{equation*}% in a bounded strictly convex domain $Ω\subset \mathbb{R}^{N}$. Here, $h $ is a strictly convex function with power-like growth $h(s)\sim s^{q}$ for $% q \in (1,2]$, and the weights $a(x), b(x)$ exhibit prescribed singular growth near $\partial Ω$. Our main contributions are threefold: (i) we prove existence and uniqueness via Perron's method and derive sharp liminf and limsup bounds for the blow-up rate $γ=(β-q+2)/(q-1)$, extending recent high-precision analytic techniques; (ii) we establish the strict convexity of solutions through the microscopic convexity principle; (iii) we provide a rigorous verification theorem identifying the solution as the value function of an infinite-horizon stochastic control problem with state constraints, in the spirit of the Lasry--Lions framework. Three distinct asymptotic regimes are identified based on the interplay between the gradient growth and the weight singularity. Numerical experiments using a monotone iterative scheme validate the theoretical boundary limits and the geometric properties of the solutions.

Figures (3)

• Figure 1: Comparison of the three boundary blow-up regimes: gradient-dominant ($q=1.6$), high-order ($q=2.5$), and critical logarithmic ($q=2.0$). The solution profiles confirm distinct asymptotic behaviors near $x = \pm 1$.
• Figure 2: Scale analysis confirming theoretical blow-up rates. Left: log-log plot for Case 1 showing slope $-\gamma = -2/3$. Right: semi-log plot for Case 3 showing linear growth in $\log(1/d)$.
• Figure 3: Theoretical consistency verification: (a) numerical solution bounded by barriers, (b) strict convexity via $u^{\prime \prime }>0$, (c) singular optimal drift structure, (d) HJB equation residual verification.

Theorems & Definitions (25)

• Theorem 1.5: Existence and Uniqueness
• Theorem 1.6: Exact Sharp Boundary Asymptotics
• Lemma 2.1: Convex defining function
• proof
• Lemma 2.2: Explicit Barriers
• proof
• Remark 2.3: Role of strict convexity
• Lemma 3.1: Comparison principle
• proof
• proof : Proof of Theorem \ref{['thm:1']}