Existence of positive solutions to a Laplace equation with nonlinear boundary condition

The positive solutions of a Laplace equation with a superlinear nonlinear boundary condition on a bounded domain are studied. For higher-dimensional domains, it is shown that non-constant positive solutions bifurcate from a branch of trivial solutions at a sequence of bifurcation points, and under a...

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Bibliographic Details
Published in:Zeitschrift für angewandte Mathematik und Physik 2015-12, Vol.66 (6), p.3061-3083
Main Authors: Kim, C.-G., Liang, Z.-P., Shi, J.-P.
Format: Article
Language:English
Subjects:
Bifurcations
Boundary conditions
Engineering
Laplace equation
Mathematical analysis
Mathematical Methods in Physics
Nonlinearity
Sequences
Theoretical and Applied Mechanics
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Summary:The positive solutions of a Laplace equation with a superlinear nonlinear boundary condition on a bounded domain are studied. For higher-dimensional domains, it is shown that non-constant positive solutions bifurcate from a branch of trivial solutions at a sequence of bifurcation points, and under additional conditions on nonlinearity, the existence of a non-constant positive solution for any sufficiently large parameter value is proved by using variational approach. It is also proved that for one-dimensional domain, there is only one bifurcation point, all non-constant positive solutions lie on the bifurcating curve, and for large parameter values, there exist at least two non-constant positive solutions. For a special case, there are exactly two non-constant positive solutions.
ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-015-0578-y