Bubbles with prescribed mean curvature: The variational approach

Let H:^3→ R be a C^1 mapping such that H(p)→H_∞>0 as |p|→∞. We show that when H satisfies some global conditions then there exists an H-bubble, namely a sphere S in R^3 such that the mean curvature of S at any regular point p∈S equals H(p)