Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation, Volume 168, Issue 798We are concerned with the nonnegative solutions of $\Delta u = u^2$ 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]. 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 $\Delta u = u^2$ in $D$ is the increasing limit of moderate solutions. |
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Perspectives in Nonlinear Partial Differential Equations: In Honor of Haïm ... Henri Berestycki No preview available - 2007 |