Bubbles with prescribed mean curvature: The variational approach

Let H:R3→RH:R3→R be a C1C1 mapping such that H(p)→H∞>0H(p)→H∞>0 as ∣p∣→∞∣p∣→∞. We show that when HH satisfies some global conditions then there exists an HH-bubble, namely a sphere SS in R3R3 such that the mean curvature of SS at any regular point p∈Sp∈S equals H(p)H(p).