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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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