Classical and non-classical solutions of a prescribed curvature equation

Bonheure, Denis;Habets, Patrick;Obersnel, Franco;Omari, Pierpaolo
(2007) Journal of Differential Equations — Vol. 243, n° 2, p. 208-237 (2007)

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Authors
  • Bonheure, DenisUCLouvain
    Author
  • Habets, PatrickUCLouvain
    Author
  • Obersnel, Franco
    Author
  • Omari, Pierpaolo
    Author
Abstract
We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem -(u'/root 1+u'(2))' = lambda f(t, u), u(0) = 0, u(1) = 0, depending on the behaviour at the origin and at infinity of the potential integral(u)(0) f(t, s)ds. Besides solutions in W-2,W-1 (0, 1), we also consider solutions in W-loc(2,1) (0, 1) which are possibly discontinuous at the endpoints of [0, 1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem. (C) 2007 Elsevier Inc. All rights reserved.
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Bonheure, D., Habets, P., Obersnel, F., & Omari, P. (2007). Classical and non-classical solutions of a prescribed curvature equation. Journal of Differential Equations, 243(2), 208-237. https://doi.org/10.1016/j.jde.2007.05.031 (Original work published 2007)