Uniform bounds for higher-order semilinear problems in conformal dimension

Abstract

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-Δ)^m u=h(x,u)\quad&\mbox{in }Ω,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partialΩ, \end{cases} \end{equation} 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. The analogue 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.

Minor corrections

References in corpus (3)

• Concentration-compactness phenomena in the higher order Liouville's equation
• Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results
• A-priori bounds for a quasilinear problem in critical dimension

Cited by in corpus (1)

• A-priori bounds for a quasilinear problem in critical dimension