A note on the existence of H-bubbles via perturbation methods

We study the problem of existence of surfaces in ${\mathbf{R}}^3$ parametrized on the sphere ${\mathbb{S}}^2$ with prescribed mean curvature $H$ in the perturbative case, i.e. for $H=H_0+ϵH_1$, where $H_0$ is a nonzero constant, $H_1$ is a $C^2$ function and $ϵ$ is a small perturbation parameter.