Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

We investigate the following problem $$\begin{aligned} -\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta }*\frac{|u|^{b}}{|x|^{\alpha }}\right) \frac{|u|^{b-2}}{|x|^{\alpha }}u+\lambda \left( |x|^{-\gamma }*\frac{|u|^{c}}{|x|^{\beta }}\right) \frac{|u|^{c-2}}{|x|^{\beta }}u \quad \text { in }{\mathbb {R}}^{N}, \end{aligned}$$ - div ( v ( x ) | ∇ u | m - 2 ∇ u ) + V ( x ) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where $$b, c, \alpha , \beta >0$$ b , c , α , β > 0 , $$\theta ,\gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$N\ge 3$$ N ≥ 3 , $$2\le m< \infty$$ 2 ≤ m < ∞ and $$\lambda \in {\mathbb {R}}$$ λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.