Classical solutions up to the boundary to some higher order semilinear Dirichlet problems

We consider the semilinear Dirichlet problem (-Delta)(m)u + g(., u) = f in bounded domains Omega subset of R-n under homogeneous Dirichlet boundary conditions (partial derivative/partial derivative v)(i)u = 0 for i = 0, ..., m - 1 on the smooth boundary partial derivative Omega. We assume that g fulfils a sign condition u g(x, u) >= 0 for all (x, u) is an element of Omega x R, which gives the coercivity of the corresponding variational formulation. For the second order problem, i.e. m = 1, with this sign condition on g the maximum principle will then imply that such a solution is classical no matter the growth of g. For higher order problems the existence of a classical solution is not obvious, unless g has a growth rate below the critical value for a Sobolev imbedding. Grunau and Sweers (1997), proved that there is a solution that is classical in the interior, allowing a one-sided larger growth rate, when compensated by a lower growth rate on the other side. We improve this result by showing the existence of a solution, that is classical up to the boundary, i.e. u is an element of C-2m,C-gamma((Omega) over bar) boolean AND C-0(m-1) ((Omega) over bar). For this we use recent estimates for the Green function of the polyharmonic Dirichlet problem and an approximation procedure. (C) 2021 Elsevier Ltd. All rights reserved