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

Abstract

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