A Double Phase Problem Involving Hardy Potentials

In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} \partial , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ is an open, bounded set Lipschitz boundary, $$0\in $$ $$N\ge 2$$ $$1<p<q<N$$ weight $$a(\cdot )\ge 0$$ $$\gamma a real parameter and f subcritical function. By variational method, provide existence of non-trivial weak solution on Musielak-Orlicz-Sobolev space $$W^{1,{\mathcal {H}}}_0(\Omega )$$ modular function $${\mathcal {H}}(t,x)=t^p+a(x)t^q$$ . For this, first introduce Hardy inequalities for under suitable assumptions