LYAPUNOV-TYPE INEQUALITY FOR EXTREMAL PUCCI’S EQUATIONS

Abstract

AbstractIn this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.