Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases

Alessio Pomponio

doi:10.3934/dcds.2018169

Article Contents

2018, Volume 38, Issue 8: 3899-3911. Doi: 10.3934/dcds.2018169

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Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases

Alessio Pomponio

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

Received: August 31, 2017

Revised: February 28, 2018

Published: May 2018

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Abstract

This paper deals with the prescribed mean curvature equations

$ - {\text{div}}\left( {\frac{{\nabla u}}{{\sqrt {1 \pm |\nabla u{|^2}} }}} \right) = g(u){\text{ }}\;\;\;\;{\text{in }}{\mathbb{R}^N},$

both in the Euclidean case, with the sign "+", and in the Lorentz-Minkowski case, with the sign "-", for N ≥ 1 under the assumption g'(0)>0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N≥ 2.

Keywords:

Prescribed mean curvature equations,
oscillating solutions.

Mathematics Subject Classification: 35B05, 35J93.

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