On uniqueness of a solution of \(Lu=u^\alpha\) with given trace (Q1583632)

A boundary trace \((\Gamma, \nu)\) of a solution of \(\Delta u = u^a\) in a bounded smooth domain in \(\mathbb{R}^d\) was first constructed by Le Gall 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, Veron, Dynkin and Kuznetsov gave analytic and probabilistic generalization of the concept of trace to the case of arbitrary \(\alpha > 1, d \geq 1\). However, it was shown by Le Gall that the trace, in general, does not define a solution uniquely in case \(d \geq(\alpha +1)/(\alpha -1)\). 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.