Question about Lagrange Multiplier and Dot Product of Local Cauchy Problem

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I would have two questions from the book "Semilinear Schrödinger Equations" by Cazenave. The first is:

  • Wouldn't the Lagrange multiplier span also over $-\Delta\varphi$?

The other question is:

  • There is this part of taking the scalar product of the equation, but I completly can't follow how this is computed. Any idea?

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