Existence of ground state solutions for a Choquard double phase problem

TL;DR

This paper proves the existence of ground state solutions for a class of quasilinear elliptic equations involving a double phase operator and a Choquard term, using variational methods and the Hardy-Littlewood-Sobolev inequality.

Contribution

It establishes the existence of ground state solutions for a novel class of double phase Choquard problems with subcritical growth, expanding the understanding of such nonlocal equations.

Findings

Existence of ground state solutions under various conditions.

Application of Hardy-Littlewood-Sobolev inequality and variational methods.

Extension to equations with double phase operators and nonlocal terms.

Abstract

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.