Given a C^1 function H: R^3 → R, we look for H-bubbles, i.e., surfaces in R^3 parametrized by the sphere S^2 with mean curvature H at every regular point. Here we study the case H(u)=H_0(u)+∈H_1(u) where H_0 is some "good" curvature (for which there exist H_ 0-bubbles with minimal energy, uniformly bounded in L^∞), ∈ is the smallness parameter, and H_1 is any C^1 function
