H-bubbles in a perturbative setting: The finite-dimensional reduction method

Given a regular function H: R^3 → R, we look for H-bubbles, that is, regular surfaces in R^3 parametrized on the sphere S^2, with mean curvature H at every point. Here we study the case of H(u) = H_0 + εH_1(u) =: H_ε(u), where H_0 is a nonzero constant, ε is the smallness parameter, and H_1 is any C^2-function. We prove that if p̄ ∈ R^3 is a "good" stationary point for a suitable Melnikov-type function Γ, then for |ε| small there exists an H_ε-bubble ω_ε that converges to a sphere of radius 1/ |H_0| centered at at p̄, as ε → 0.