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


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.


Weak Solution Neumann Condition Positive Weak Solution Nonradial Solution 
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© Springer Science+Business Media, Inc. 2005

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  • A. P. Scheglova

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