In this paper, we consider the following p-Laplacian problem with Hardy potential: \(\begin{aligned} \begin{aligned} -\text {div}(a(y)|\nabla v|^{p-2} \nabla v) + b(y) |v|^{p-2} v + \mu \frac{v^s}{|y|^p}&= f(y,v) \ \text {in} \ \Omega ,\\ v&> 0 \ \text {in} \ \Omega , \\ v&= 0 \ \text {on} \ \partial \Omega , \end{aligned} \end{aligned}\) here \(p \in (1,n), \ \Omega \ (\subset {\mathbb {R}}^n)\) is an exterior domain. We assume that the function f has either superlinear or sublinear growth with respect to the variable v. By using critical point theory, we establish the existence of a weak solution to this problem.
