The Dirichlet Problem for Some Semilinear Elliptic Differential Equations of Arbitrary Order
Received: November 4, 1990 Abstract. For some special semilinear elliptic equations, e.g. (-A)u + g(u) = f, the existence of a solution of the Dirichlet problem is shown, which is classical in the interior of the domain and takes on the boundary values in a weak sense. The nonlinearity g : E -» E has to satisfy the sign condition t · g(t) S 0 and to be bounded from below, e.g. g(t) = exp(t) -1. Furthermore it is possible to treat nonlinearities g with growth conditions, which can be viewed as inteipolation between the above extreme case and the standard one lg(t)l < C (l + ltl"). The crucial lemma is a local maximum principle for solutions of differential inequalities (-A)u S f, which makes use of the positivity of Green's function for (~A) in balls.