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Nonlinear boundary value problems involving the extrinsic mean curvature operator. (English) Zbl 1340.35092
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.

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
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