Uniform bounds for higher-order semilinear problems in conformal dimension. (March 2020)

• Record Type: Journal Article
• Authors: Mancini, Gabriele; Romani, Giulio
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: primary 35J40 35B45 -- secondary 35J91 35B44; Higher-order elliptic problems -- A-priori estimates -- Positive solutions -- Blow-up; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111717
• Languages: English
• ISSNs: 0362-546X

Abstract

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.