# Nonlinear boundary value problems involving the extrinsic mean curvature operator

Mathematica Bohemica (2014)

• Volume: 139, Issue: 2, page 299-313
• ISSN: 0862-7959

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## Abstract

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The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $\nabla ·\left(\frac{\nabla v}{\sqrt{1-{|\nabla v|}^{2}}}\right)=f\left(|x|,v\right)\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{B}_{R},\phantom{\rule{1.0em}{0ex}}u=0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial {B}_{R},$ where ${B}_{R}$ is the open ball of center $0$ and radius $R$ in ${ℝ}^{n}$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.

## How to cite

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Mawhin, Jean. "Nonlinear boundary value problems involving the extrinsic mean curvature operator." Mathematica Bohemica 139.2 (2014): 299-313. <http://eudml.org/doc/261917>.

@article{Mawhin2014,
abstract = {The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $\nabla \cdot \bigg (\frac\{\nabla v\}\{\sqrt\{1 - |\nabla v|^2\}\}\bigg ) = f(|x|,v) \quad \text\{in\} \ B\_R,\quad u = 0 \quad \text\{on\} \ \partial B\_R ,$ where $B_R$ is the open ball of center $0$ and radius $R$ in $\mathbb \{R\}^n$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.},
author = {Mawhin, Jean},
journal = {Mathematica Bohemica},
keywords = {extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory; extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory},
language = {eng},
number = {2},
pages = {299-313},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonlinear boundary value problems involving the extrinsic mean curvature operator},
url = {http://eudml.org/doc/261917},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Mawhin, Jean
TI - Nonlinear boundary value problems involving the extrinsic mean curvature operator
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 299
EP - 313
AB - The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $\nabla \cdot \bigg (\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}}\bigg ) = f(|x|,v) \quad \text{in} \ B_R,\quad u = 0 \quad \text{on} \ \partial B_R ,$ where $B_R$ is the open ball of center $0$ and radius $R$ in $\mathbb {R}^n$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.
LA - eng
KW - extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory; extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory
UR - http://eudml.org/doc/261917
ER -

## References

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