Nonlinear boundary value problems involving the extrinsic mean curvature operator

Jean Mawhin

Mathematica Bohemica (2014)

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

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 ∇ · ∇ v 1 - | ∇ v | 2 = f ( | x | , v ) in B R , u = 0 on ∂ 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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