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Journal of Mathematical Sciences

, Volume 128, Issue 5, pp 3306–3333 | Cite as

Multiplicity of Solutions to a Boundary-Value Problem with Nonlinear Neumann Condition

  • A. P. Scheglova
Article

Abstract

We consider the boundary-value problem
$$- \Delta _p u + \left| u \right|^{p - 2} u = 0\quad in\;B_R ,$$
$$\left| {\nabla u} \right|^{p - 2} \left\langle {\nabla u;n} \right\rangle = \left| u \right|^{q - 2} u\quad on\;S_R ,$$
where \(\Delta _p u = div(\left| {\nabla u} \right|^{p - 2} \nabla u)\) and n is the unit outward normal. We show that there exist so many nonequivalent positive weak solutions as prescribed under certain conditions on q and R. We construct nonradial solutions for [(n + 1)/2] + 1 ⩽ p < n and some q. Bibliography: 18 titles.

Keywords

Weak Solution Neumann Condition Positive Weak Solution Nonradial Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, Inc. 2005

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  • A. P. Scheglova

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