Weak solutions to complex Hessian equations in the class Nᵃₘ(Ω,ω,ϕ) in a ball of Cⁿ

In this paper, we study weak solutions to complex Hessian equations of the form (ω+ ddᶜ φ)ᵐβⁿ⁻ᵐ= F(φ,.)dμ in the class Nᵃₘ(Ω,ω,ϕ) on a ball in Cⁿ, where $ω$ is a real smooth $(1,1)$-form, $μ$ is a positive measure which puts no mass on m-polar subsets, $Ω$ is a ball in Cⁿ and F(t,z): R×Ω [0,+∞) is a upper semi-continuous function on R×Ω, 0≤ F(t,z)≤ G(z) for all (t,z)× Ω, G(z)∈ L¹loc(Ω, dμ) and $F(t,z)$ is continuous in the first variable. Our result extends a result of Salouf in {S23} in which he solved Monge-Ampère type equations (ω+ ddᶜ φ)ⁿ= F(φ,.)dμ with $F(t,z)$ is a continuous non-decreasing function in the first variable function and φ(Ω).