Mawhin, Jean Nonlinear boundary value problems involving the extrinsic mean curvature operator. (English) Zbl 1340.35092 Math. Bohem. 139, No. 2, 299-313 (2014). Summary: 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. Cited in 5 Documents MSC: 35J93 Quasilinear elliptic equations with mean curvature operator 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35B09 Positive solutions to PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B07 Axially symmetric solutions to PDEs Keywords:extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory PDFBibTeX XMLCite \textit{J. Mawhin}, Math. Bohem. 139, No. 2, 299--313 (2014; Zbl 1340.35092) Full Text: Link