We present CO(3-2), CO(2-1), and 230 GHz (1.3 mm) continuum images of nearby galaxies taken with the Submillimeter Array (SMA). Our main topic is to study the relation between higher-J molecular gas (e.g., CO J=3-2, 2-1) and nuclear activities (e.g., active galactic nuclei [AGNs] and starbursts). The nearby Seyfert 2 galaxy M51 shows strong CO(3-2) emission from the circumnuclear molecular gas, with an intensity twice as strong as that of the CO(1-0) emission. Strong CO(3-2) emission enhancement suggests that the circum nuclear molecular gas in M51 is warm and dense, which may be related to the AGN activities. Molecular gas in the nearby moderate starburst galaxy NGC 6946 is distributed along the large-scale bar or spiral arms and along the minibar, and the multi-J CO line images show very similar distribution to each other. For this galaxy, there is no clear enhancement in higher-J lines as seen in M51, which may be because NGC 6946 does not have clear AGN activities. Based on the results of these two galaxies, the physical conditions of the circum nuclear molecular gas may be related to the AGN activities. We also observed the nearby edge-on starburst galaxy NGC 3628 and the starburst/Seyfert composite galaxy NGC 4945 with the CO(2-1) line and 230 GHz (1.3 mm) continuum emission. These information will give us some hints for understanding the relation between nuclear activities and circum nuclear molecular gas and dust.