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We investigate two dark energy cosmological models (i.e., the $\Lambda$CDMand $\phi$CDM models) with massive neutrinos assuming two different neutrinomass hierarchies in both the spatially flat and non-flat scenarios, where inthe $\phi$CDM model the scalar field possesses an inverse power-law potential,$V(\phi)\propto {\phi}^{-\alpha}$ ($\alpha>0$). Cosmic microwave backgrounddata from Planck 2015, baryon acoustic oscillations data from 6dFGS, SDSS-MGS,BOSS-LOWZ and BOSS CMASS-DR11, the JLA compilation of Type Ia supernovaapparent magnitude observations, and the Hubble Space Telescope $H_0$ prior,are jointly employed to constrain the model parameters. We first determineconstraints assuming three species of degenerate massive neutrinos. In thespatially flat (non-flat) $\Lambda$CDM model, the sum of neutrino masses isbounded as $\Sigma m_{\nu} < 0.165 (0.299)$ eV at 95% confidence level (CL).Correspondingly, in the flat (non-flat) $\phi$CDM model, we find $\Sigmam_{\nu} < 0.164 (0.301)$ eV at 95% CL. The inclusion of spatial curvature as afree parameter results in a significant broadening of confidence regions for$\Sigma m_{\nu}$ and other parameters. In the scenario where the total neutrinomass is dominated by the heaviest neutrino mass eigenstate, we can obtain thesimilar conclusions as those obtained in the degenerate neutrino mass scenario.In addition, the results show that the bounds on $\Sigma m_{\nu}$ based on twodifferent neutrino mass hierarchies have insignificant differences in thespatially flat case for both the $\Lambda$CDM and $\phi$CDM models, however,the corresponding differences are larger in the non-flat case.

Constraints on non-flat cosmologies with massive neutrinos after Planck 2015