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In the present paper the Dirichlet problem for semilinear elliptic and parabolic equa- tions in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u| p1 u, p>1, is formulated. For sublinear case (for example ln(1+|u|) or |u| p1 u, p < 1) this condition is automatically fulfilled. Our approach gives new a priori estimate of the solution for superlinear, sublinear and linear case as well.

The Dirichlet problem for second order semilinear elliptic and parabolic equations