In this paper we investigate the existence of solution for the following nonlocal problem with Stein-Weiss convolution term \(-\Delta_{\Phi}u+V(x)\phi(|u|)u=\dfrac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}|y|^\alpha}{\rm d}y\right)K(x)f(u(x)),\;\;x\in \mathbb{R}^{N},\) where α ≥ 0, N ≥ 2, λ ≥ 0 is a positive parameter, V, K ∈ C(ℝN, [0, ∞)) are nonnegative functions that may vanish at infinity, the function f ∈ C(ℝ, ℝ) is quasicritical and \(F(t)=\int_{0}^{t}f(s){\rm d}s\) . To establish our existence and regularity results, we use the Hardy-type inequalities for Orlicz-Sobolev Space and the Stein-Weiss inequality together with a varia-tional technique based on the mountain pass theorem for a functional that is not necessarily in C1. Furthermore, we also prove the existence of a ground state solution by the method of Nehari manifold in the case where the strict monotonicity condition on f is not required. This work incorporates the case where the \(\mathcal{N}\) -function \(\tilde{\Phi}\) does not verify the Δ2-condition.
