A variational Approach to complex Hessian equations in ${\mathbb{C}}^n$

Let $\mathrm{\Omega }$ be a $m$-hyperconvex domain of ${\mathbb{C}}^n$ and $\beta$ be the standard K"{a}hler form in ${\mathbb{C}}^n$. We introduce finite energy classes of $m$-subharmonic functions of Cegrell type, ${\mathcal{E}}_m^p,p>0$ and ${\mathcal{F}}_m$. Using a variational method we show that the degenerate complex Hessian equation $(d{d}^c\phi {)}^m\wedge {\beta }^{n-m}=\mu$ has a unique solution in ${\mathcal{E}}_m^1$ if and only if every function in ${\mathcal{E}}_m^1$ is integrable with respect to $\mu$. If $\mu$ has finite total mass and does not charge $m$-polar sets, then the equation has a unique solution in ${\mathcal{F}}_m$.