Geodesic
Nonlinear boundary value problems involving the extrinsic mean curvature operator
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 299-313

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The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $$ \nabla \cdot \bigg (\frac {\nabla v}{\sqrt {1 - |\nabla v|^2}}\bigg ) = f(|x|,v) \quad \text {in} \ B_R,\quad u = 0 \quad \text {on} \ \partial B_R , $$ 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.
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.
DOI : 10.21136/MB.2014.143856
Classification : 35-02, 35A16, 35B09, 35B38, 35J20, 35J25, 35J60, 35J87, 35J93
Keywords: extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory 
Mawhin, Jean. Nonlinear boundary value problems involving the extrinsic mean curvature operator. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 299-313. doi: 10.21136/MB.2014.143856
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