Classification of global solutions to the singular equation $-\Delta u=f(X)\cdot u^{-\gamma }$ in a Lipschitz epigraphical cone

Abstract

We construct and classify all global solutions to the singular equation $$-\Delta u=f(X)\cdot u^{-\gamma }$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<\lambda \le f(X)\le \Lambda$. The existence and non-existence of a global solution is solely determined by the exponent $\gamma$ of the equation and the ``frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the ``asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an ``oscillation reduction" argument on the ``asymptotic slope".