Some existence results for a nonlocal non-isotropic problem

In this paper we deal with the following problem [formula] where $\Omega$ is a bounded regular domain in $ \mathbb{R}^{n}$. We will assume without loss of generality that $1 \leq p_{1} \leq p_{2} \leq \ldots \leq p_{N}$ and that ƒ and g are non-negative functions belonging to a suitable Lebesgue space $L^{m}(\Omega), 1 0, b > 0 \text{ and } 0 < \gamma < 1$