Classical and non-classical solutions of a prescribed curvature equation
By Denis Bonheure, P. Habets, Franco Obersnel, Pierpaolo Omari. 2007
113 citations recorded by the source corpus.
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.
View Paper: https://www.semanticscholar.org/paper/5f72b26d3eb65a7339854593a07cf236403e9824
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- Denis Bonheure
- P. Habets
- Franco Obersnel
Citations in the corpus
- Classical and non-Classical Positive Solutions of a Prescribed Curvature Equation with Singularities (2007, background): “In [2] existence and multiplicity of positive solutions of (1) have been established under various types of assumptions on the behaviour of the function f at zero and at infinity.”
- Positive solutions of the Dirichlet problem for the prescribed mean curvature equation (2010, background): “Unlike the onedimensional case we discussed in [5], we are not able here to prove the existence of a positive solution for each λ > 0.”
- Uniqueness (2021, unknown): “We remark that the existence of small C 1 -solutions was proved in the one-dimensional case in [20, 44] and the existence of small solutions in higher dimensions in [67].”
- Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space (2013, unknown): “Some specific configurations of the limits of F (x, s)/s(2) at 0 and of F (x, s)/s at +∞ will then yield the existence of one, two, three, or infinitely many positive solutions of (1), thus reproducing the multiplicity phenomena already pointed out in [5] in the one-dimensional case.”