A double phase problem involving Hardy potentials

Abstract

In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)= γ\left(\displaystyle\frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle\frac{|u|^{q-2}u}{|x|^q}\right)+f(x,u) & \mbox{in } Ω,\\ u=0 & \mbox{in } \partialΩ, \end{array} \right. $$ 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 $W_0^{1,\mathcal{H}}(\Omega )$, with modular function $\mathcal{H}(t,x)=t^p+a(x)t^q$. For this, we first introduce the Hardy inequalities for space $W_0^{1,\mathcal{H}}(\Omega )$, under suitable assumptions on $a(\cdot )$.