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For the boundary value problem      $$ - \Delta _p u = \left| x \right|^\alpha u^{q - 1} in B_1 , u|_{\partial B_1 } = 0$$       and some more general problems the existence of any a priori given number of different positive solutions is established under suitable conditions on q and α. Nonradial solutions to the problem are constructed for some supercritical q. Bibliography: 30 titles.

Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation