Uniform bounds for higher-order semilinear problems in conformal dimension

TL;DR

This paper derives uniform bounds for solutions of higher-order semilinear elliptic problems in conformal dimension, leading to existence results via degree theory, applicable to various boundary conditions and domain types.

Contribution

It provides the first uniform a-priori estimates for higher-order semilinear problems in conformal dimension, extending to Navier boundary conditions and different domain geometries.

Findings

Established uniform a-priori bounds for solutions.

Proved existence of positive solutions using degree theory.

Extended results to Navier boundary conditions.

Abstract

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