On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace

Abstract

A boundary trace $(\Gamma, \nu)$ of a solution of $\Delta u = u^\alpha$ in a bounded smooth domain in $\mathbb{R}^d$ was first constructed by Le Gall \cite{LGOne} who described all possible traces for $\alpha = 2, d= 2$ in which case a solution is defined uniquely by its trace. In a number of publications, Marcus, V\'eron, Dynkin and Kuznetsov gave analytic and probabilistic generalization of the concept of trace to the case of arbitrary $\alpha > 1, d \ge 1$. However, it was shown by Le Gall  that the trace, in general, does not define a solution uniquely in case $d\ge (\alpha +1)/(\alpha -1)$. He offered a sufficient condition for the uniqueness and conjectured that a uniqueness should be valid if the singular part $\Gamma$ of the trace coincides with the set of all explosion points of the measure $\nu$. Here, we establish a necessary condition for the uniqueness which implies a negative answer to the above conjecture.