An upper bound for positive solutions of the equation $\Delta u=u^\alpha$

In 2002 Mselati proved that every positive solution of the equation $\Delta u=u^2$ in a bounded domain of class $C^4$ 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 $\partial E$. 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 $\Delta u=u^\alpha$ with $1<\alpha \le 2$. This is a part of the program: to classify all positive solutions of this equation.