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research
Uniform bounds for higher-order semilinear problems in conformal dimension
Authors
Denis Bourguet
Brad S. Coates
+5Â more
Kanglai He
Kyung Seok Kim
Jing Li
Sergine Ponsard
Zhenying Wang
Publication date
14 August 2014
Publisher
Doi
Cite
View
on
arXiv
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
h
h
is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when
Ω
\Omega
Ω
is a ball or, provided an energy control on solutions is prescribed, when
Ω
\Omega
Ω
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.Comment: Minor correction
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Last time updated on 30/10/2019