Existence of H-bubbles in a perturbative setting

Given a $C^{1}$ function $H\colon\mathbb{R}^{3}\to \mathbb{R}$, we look for $H$-bubbles, i.e, surfaces in $\mathbb{R}^{3}$ parametrized by the sphere $\mathbb{S}^{2}$ with mean curvature $H$ at every regular point. Here we study the case $H(u)=H_{0}(u)+ \varepsilon H_{1}(u)$ where $H_{0}$ is some "good" curvature (for which there exist $H_{0}$-bubbles with minimal energy, uniformly bounded in $L^{\infty}$), $\varepsilon$ is the smallness parameter, and $H_{1}$ is {\em any} $C^{1}$ function.