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In this paper, we consider the isoperimetric problem in the space  with a density. Our result states that, if the density  is lower semi-continuous and converges to a limit  at infinity, with  far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331–365, .

Existence of Isoperimetric Sets with Densities “Converging from Below” on $${\mathbb {R}}^N$$ R N