Ljusternik-Schnirelman Theory and the Existence of Several Eigenvectors

In Chapter 43 we proved the existence of an eigenvector. Now we concern ourselves with the eigenvalue problem 1<m:math display='block'> <m:mrow> <m:mi>A</m:mi><m:mi>u</m:mi><m:mo>=</m:mo><m:mi>&#x03BB;</m:mi><m:mi>B</m:mi><m:mi>u</m:mi><m:mo></m:mo><m:mi>u</m:mi><m:mo>&#x2208;</m:mo><m:mi>X</m:mi><m:mo></m:mo><m:mi>&#x03BB;</m:mi><m:mo>&#x2208;</m:mo><m:mi>&#x211D;</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$Au = \lambda Bu,u \in X,\lambda \in \mathbb{R}$ and we will prove the existence of several eigenvectors for (1) within the generalized context of the Courant maximum-minimum principle. In this connection, in an essential way, we use the fact that A and B are odd potential operators, i.e., A = F′, B = G′, and A(− μ) = − A(μ), B(−μ) = − B(μ) for all μ ∈ X. We have already explained the basic idea of the Ljusternik-Schnirelman theory in Section 37.26, and we recommend that the reader first study Section 37.26 again.