In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{ align*} -\mathcal{ L}_{ p,q}^{ a}(u) + |u|^{ p-2}u+ a(x) |u|^{ q-2}u = \left( \int_{ \mathbb{ R}^N} \frac{ F(y, u)}{ |x-y|^\mu}\,\mathrm{ d} y\right)f(x,u) \quad\text{ in } \mathbb{ R}^N, \end{ align*} where $\mathcal{ L}_{ p,q}^{ a}$ is the double phase operator given by \begin{ align*} \mathcal{ L}_{ p,q}^{ a}(u):= \operatorname{ div}\big(|\nabla u|^{ p-2}\nabla u + a(x) |\nabla u|^{ q-2}\nabla u \big), \quad u\in W^{ 1,\mathcal{ H}}(\mathbb{ R}^N), \end{ align*} $0<\mu<N$, $1<p<N$, $p<q<p+ \frac{ \alpha p}{ N}$, $0 \leq a(\cdot)\in C^{ 0,\alpha}(\mathbb{ R}^N)$ with $\alpha \in (0,1]$ and $f\colon\mathbb{ R}^N\times\mathbb{ R}\to\mathbb{ R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.
Categories: math.AP