Abstract
In the previous chapter we noted that in our study of semilinear elliptic partial differential equations of the form
in unbounded domains, we required that the resolvent set of \(\mathcal A\) not be empty. For convenience, we assumed that it contain 0. This allowed us to choose an interval \((a,b) \subset \rho (\mathcal A),\) where a < 0 < b. We let \( D = D(|\mathcal A|{ }^{(1/2)}).\) With the scalar product \((u,v)_D= (|\mathcal A|{ }^{(1/2)}u,|\mathcal A|{ }^{(1/2)}v),\) it became a Hilbert space. We let
be the negative and positive invariant subspaces of \(\mathcal A.\) Then
and
The hypotheses of our theorems depended on a and b using the fact that 0 was embedded in \(\rho (\mathcal A).\) The purpose of the present chapter is to study the situation when 0 is a boundary point of \(\rho (\mathcal A)\) and the arguments do not work.
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Schechter, M. (2020). Zero in the Spectrum. In: Critical Point Theory. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-45603-0_8
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DOI: https://doi.org/10.1007/978-3-030-45603-0_8
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