Uniform bounds for higher-order semilinear problems in conformal dimension. (March 2020) - Mancini, Gabriele; Romani, Giulio
Abstract: We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem ( − Δ ) m u = h ( x, u )  in  Ω, u = ∂ n u = ⋯ = ∂ n m − 1 u = 0  on  ∂ Ω, where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger–Moser–Adams inequality, either when Ω is a ball or, provided an energy control on solutions is prescribed, when Ω 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.