A variational approach to complex Hessian equations in CnCn

Let Ω be an m  -hyperconvex domain of CnCn and β   be the standard Kähler form in CnCn. We introduce finite energy classes of m  -subharmonic functions of Cegrell type, Emp(Ω), p>0p>0 and Fm(Ω)Fm(Ω). Using a variational method we show that the degenerate complex Hessian equation (ddcφ)m∧βn−m=μ(ddcφ)m∧βn−m=μ has a unique solution in Em1(Ω) if and only if every function in Em1(Ω) is integrable with respect to μ. If μ has finite total mass and does not charge m  -polar sets, then the equation has a unique solution in Fm(Ω)Fm(Ω).