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

Authors:

A. P. Scheglova

Abstract:

We consider the boundary-value problem

$- Delta _p u + left| u right|^{p - 2} u = 0quad in;B_R ,$

$left| {nabla u} right|^{p - 2} leftlangle {nabla u;n} rightrangle = left| u right|^{q - 2} uquad on;S_R ,$

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.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 121–144.

Keywords:

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