Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem {(-Delta)(m)u = h(x, u) in Omega, u = partial derivative(n)u = ... = partial derivative(m-1)(n)u = 0 on partial derivative Omega, where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when Omega is a ball or, provided an energy control on solutions is prescribed, when Omega is a smooth bounded domain. Our results are sharp within the class of distributional solutions. The analogous problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory. (C) 2019 Elsevier Ltd. All rights reserved.