This study is an open label, sequential-group, two site, multiple dose escalating study of sublingual administered asenapine in a pediatric population with schizophrenia or bipolar I disorder; in one study cohort (3a) participants with other conditions treatable with chronic antipsychotic medication can also be enrolled. Participants will receive a single sublingual placebo dose on Day -1, followed by multiple sublingual doses of asenapine twice daily (b.i.d.) for 6 days (Cohorts 1 and 2), 7 days (Cohort 3b-d), or 11 days (Cohort 3a), and a final once daily administration on Day 7 (Cohorts 1 and 2), Day 8 (Cohort 3b-d) or Day 12 (Cohort 3a).