Article
MSC:
35-02,
35A16,
35B09,
35B38,
35J20,
35J25,
35J60,
35J87,
35J93 | MR 3238841 | Zbl 06362260 | DOI: 10.21136/MB.2014.143856
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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
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.
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