It should not be hard to prove - e.g. by some minmax-characterization - that $\lambda_1$ is measurable. (As already remarked in the comments by Christian Remling, $\lambda_1$ is actually continuous, but measurability of $\lambda_1$ is sufficient for the following argument.)

Then the existence of a measurable selection $v_1$ of the zeroes (e.g. on the unit sphere) of the superposition operator generated by the Caratheodory function $f(t,v)=A(t)v-\lambda_1(t)v$ is a trivial special case of a Filippov type implicit function theorem. See, for instance, Theorem 3 in https://core.ac.uk/download/pdf/81993694.pdf.

It should not be hard to prove - e.g. by some minmax-characterization - that $\lambda_1$ is measurable. (As already remarked in the comments by Christian Remling, $\lambda_1$ is actually continuous, but measurability of $\lambda_1$ is sufficient for the following argument.)

Then the existence of a measurable selection $v_1$ of the zeroes (e.g. on the unit sphere) of the superposition operator generated by the Caratheodory function $f(t,v)=A(t)v-\lambda_1(t)v$ is a trivial special case of a Filippov type implicit function theorem. See, for instance, Theorem 3' in https://core.ac.uk/download/pdf/81993694.pdf.