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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 16, Issue 1

Issues

Corrigendum to: The Dirichlet problem with mean curvature operator in Minkowski space

Cristian Bereanu / Petru Jebelean / Jean Mawhin
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  • Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
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Published Online: 2016-01-23 | DOI: https://doi.org/10.1515/ans-2015-5030

We essentially use Lemma 2.1 which is proved in [1]. Accordingly, the set Ω must be “a bounded domain in N (N2) with boundary Ω of class C2”, instead of “an open bounded set in N with boundary Ω of class C2”. This change does not affect the validity of the results stated in the paper. But, it affects the proof of Lemma 3.1. In this view, the only necessary modification is the following.

Proof of Lemma 3.1.

It suffices to show that u0 in Ω. From (2.2), (3.4) and the integration by parts formula it follows

-Ωuv1-|u|2=Ωμ(x)|u|q-1uv-λΩg¯(x,u)v,1

for all vW1,(Ω) with v|Ω=0. We denote

𝒪:={xΩ:u(x)<0},u-:=min{0,u},𝒪:={x𝒪:|u-(x)|>0}.

From [2, Theorem A.1] we have u-W1,(Ω) and u-=u in 𝒪, u-=0N in Ω𝒪. So, taking v=u- in (1) and using hypothesis (H), it follows

-𝒪|u-|21-|u-|2=𝒪μ(x)|u-|q+10.2

If meas𝒪>0, then from (2) we get the contradiction

0>-𝒪|u-|21-|u-|20.

Consequently, meas𝒪=0 and, as u-=0N in Ω𝒪, we get that u-=0N a.e. on Ω. As Ω is a domain, we infer that u-=const., hence u-0 in Ω. This means u0 in Ω. ∎

References

  • [1]

    Corsato C., Obersnel F., Omari P. and Rivetti S., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl. 405 (2013), 227–239.  Google Scholar

  • [2]

    Kinderlehrer D. and Stampacchia G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.  Google Scholar

About the article

Published Online: 2016-01-23

Published in Print: 2016-02-01


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 173–174, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5030.

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