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A VARIATIONAL APPROACH FOR ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE PROBLEMS

Part of: Manifolds Boundary value problems

Published online by Cambridge University Press:  25 July 2014

GHASEM A. AFROUZI ,
ARMIN HADJIAN  and
GIOVANNI MOLICA BISCI
GHASEM A. AFROUZI
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran email [email protected]
ARMIN HADJIAN
Affiliation:
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, PO Box 1339, Bojnord 94531, Iran email [email protected]
GIOVANNI MOLICA BISCI*
Affiliation:
Dipartimento P.A.U., Università degli Studi “Mediterranea”, di Reggio Calabria - Feo di Vito, 89124 Reggio Calabria, Italy email [email protected]
*
✉[email protected]
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Abstract

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We discuss the multiplicity of nonnegative solutions of a parametric one-dimensional mean curvature problem. Our main effort here is to describe the configuration of the limits of a certain function, depending on the potential at zero, that yield, for certain values of the parameter, the existence of infinitely many weak nonnegative and nontrivial solutions. Moreover, thanks to a classical regularity result due to Lieberman, this sequence of solutions strongly converges to zero in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^1([0,1])$. Our approach is based on recent variational methods.

Keywords

one-dimensional prescribed curvature problemvariational methodsinfinitely many solutions

MSC classification

Secondary: 34B15: Nonlinear boundary value problems 49Q20: Variational problems in a geometric measure-theoretic setting
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 2 , October 2014 , pp. 145 - 161
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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