Does Max Planar 3-SAT admit a PTAS?
Yes, a PTAS for Max-Planar-3-SAT can be constructed by using Brenda Baker's approach.
This has been observed, for instance, in Theorem 17 in 


  Pierluigi Crescenzi and LucaTrevisan:
  "Max NP-completeness made easy"
  Theoretical Computer Science 28, (1999), Pages 65-79