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In 2002 Mselati proved that every positive solution of the equation ∆u = u2 in a bounded domain of class C4 is the limit of an increasing sequence of moderate solutions. (A solution is called moderate if it is dominated by a harmonic function.) As a part of his proof, he established an upper bound (in terms of the capacity of K) for solutions vanishing off a compact subset K of ∂E. We use a different kind of capacity (we call it the Poisson capacity) and we establish in terms of this capacity an upper bound for solutions of ∆u = uα with 1 < α ≤ 2. This is a part of the program: to classify all positive solutions of this equation.

An upper bound for positive solutions of the equation Δ𝑢=𝑢^{𝛼}