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We prove that      Δ  u + h u α  = 0 \Delta u + h{u^\alpha } = 0       has no positive solutions for certain      h h      ,      α  \alpha       by studying the linearized equation      ( Δ  + h u α  −  1 ) ψ  = e ψ  (\Delta + h{u^{\alpha - 1}})\psi = e\psi      .

Schrödinger operator methods in the study of a certain nonlinear P.D.E