The Gel'fand Problem for the Biharmonic Operator - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2013

The Gel'fand Problem for the Biharmonic Operator

Résumé

We study stable and finite Morse index solutions of the equation Δ2 = eu. If the equation is posed in ℝN, we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics of solutions at infinity. Thanks to this analysis, we prove that stable solutions of the equation on a smoothly bounded domain (supplemented with Navier boundary conditions) are smooth if and only if N ≦ 12. We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index, provided they are controlled in a suitable Morrey norm. © 2013 Springer-Verlag Berlin Heidelberg.

Dates et versions

hal-00866955 , version 1 (27-09-2013)

Identifiants

Citer

L. Dupaigne, M. Ghergu, O. Goubet, Guillaume Warnault. The Gel'fand Problem for the Biharmonic Operator. Archive for Rational Mechanics and Analysis, 2013, 208 (3), pp.725-752. ⟨10.1007/s00205-013-0613-0⟩. ⟨hal-00866955⟩
51 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More