Existence of H-bubbles in a perturbative setting

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) = H0(u) + H 1(u) where H0 is some “good” curvature (for which there exist H0-bubbles with minimal energy, uniformly bounded in L 1 ), is the smallness parameter, and H1 is any C 1 function.