We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω,u=∂nu=⋯=∂nm−1u=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.
Uniform bounds for higher-order semilinear problems in conformal dimension
Mancini G.;
2020-01-01
Abstract
We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω,u=∂nu=⋯=∂nm−1u=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.File in questo prodotto:
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