A Double Phase Problem Involving Hardy Potentials

Abstract

In this paper, we deal with the following double phase problem

$$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} &{} \text{ in } \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is an open, bounded set with Lipschitz boundary, $0\in \Omega$, $N\ge 2$, $1<p<q<N$, weight $a(\cdot )\ge 0$, $\gamma$ is a real parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space $W^{1,{\mathcal {H}}}_0(\Omega )$, with modular function ${\mathcal {H}}(t,x)=t^p+a(x)t^q$. For this, we first introduce the Hardy inequalities for space $W^{1,{\mathcal {H}}}_0(\Omega )$, under suitable assumptions on $a(\cdot )$.