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

   
@article{ECP1027,
	author = {Sergei Kuznetsov},
	title = {On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {5},
	year = {2000},
	keywords = {superdiffusion, moderate solutions, sigma-moderate solutions, stochastic boundary values, trace of a solution, explosion points.},
	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.},
	pages = {no. 15, 137-147},
	issn = {1083-589X},
	doi = {10.1214/ECP.v5-1027},    
        url = {http://ecp.ejpecp.org/article/view/1027}}