Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation - Mselati, Benoîti (Memoirs of the American Mathematical Society, 2004)
We are concerned with the nonnegative solutions of $Δ u = u^2$ in a bounded and smooth domain in $\mathbb{R}^d$. We prove that they are uniquely determined by their fine trace on the boundary as defined in [DK98a], thus answering a major open question of [Dy02]. In this title, a probabilistic formula for a solution in terms of its fine trace and of the Brownian snake is also provided. A major role is played by the solutions which are dominated by a harmonic function in $D$. The latters are called moderate in Dynkin's terminology. We show that every nonnegative solution of $Δ u = u^2$ in $D$ is the increasing limit of moderate solutions.