We essentially use Lemma 2.1 which is proved in [1]. Accordingly, the set Ω must be “a bounded *domain* in ${\mathbb{R}}^{N}$ ($N\ge 2$) with boundary $\partial \mathrm{\Omega}$ of class ${C}^{2}$”, instead of “an *open bounded set* in ${\mathbb{R}}^{N}$ with boundary $\partial \mathrm{\Omega}$ of class ${C}^{2}$”. 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 $u\ge 0$ in Ω. From (2.2), (3.4) and the integration by parts formula it follows

$-{\int}_{\mathrm{\Omega}}\frac{\nabla u\cdot \nabla v}{\sqrt{1-{|\nabla u|}^{2}}}={\int}_{\mathrm{\Omega}}\mu (x){|u|}^{q-1}uv-\lambda {\int}_{\mathrm{\Omega}}\overline{g}(x,u)v,$1

for all $v\in {W}^{1,\mathrm{\infty}}(\mathrm{\Omega})$ with ${v|}_{\partial \mathrm{\Omega}}=0$. We denote

$\mathcal{\mathcal{O}}:=\{x\in \mathrm{\Omega}:u(x)<0\},{u}^{-}:=\mathrm{min}\{0,u\},{\mathcal{\mathcal{O}}}^{\prime}:=\{x\in \mathcal{\mathcal{O}}:|\nabla {u}^{-}(x)|>0\}.$

From [2, Theorem A.1] we have ${u}^{-}\in {W}^{1,\mathrm{\infty}}(\mathrm{\Omega})$ and $\nabla {u}^{-}=\nabla u$ in $\mathcal{\mathcal{O}}$, $\nabla {u}^{-}={0}_{{\mathbb{R}}^{N}}$ in $\mathrm{\Omega}\setminus \mathcal{\mathcal{O}}$.
So, taking $v={u}^{-}$ in (1) and using hypothesis (*H*), it follows

$-{\int}_{\mathcal{\mathcal{O}}}\frac{{|\nabla {u}^{-}|}^{2}}{\sqrt{1-{|\nabla {u}^{-}|}^{2}}}={\int}_{\mathcal{\mathcal{O}}}\mu (x){|{u}^{-}|}^{q+1}\ge 0.$2

If $meas{\mathcal{\mathcal{O}}}^{\prime}>0$, then from (2) we get the contradiction

$0>-{\int}_{{\mathcal{\mathcal{O}}}^{\prime}}\frac{{|\nabla {u}^{-}|}^{2}}{\sqrt{1-{|\nabla {u}^{-}|}^{2}}}\ge 0.$

Consequently, $meas{\mathcal{\mathcal{O}}}^{\prime}=0$ and, as $\nabla {u}^{-}={0}_{{\mathbb{R}}^{N}}$ in $\mathrm{\Omega}\setminus \mathcal{\mathcal{O}}$, we get that $\nabla {u}^{-}={0}_{{\mathbb{R}}^{N}}$ a.e. on Ω. As Ω is a domain, we infer that ${u}^{-}=\text{const.}$, hence ${u}^{-}\equiv 0$ in Ω. This means $u\ge 0$ 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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