Ground state solutions for non-autonomous fractional Choquard equations

TL;DR

This paper proves the existence of ground state solutions for a class of non-autonomous fractional Choquard equations with variable coefficients, extending the understanding of such equations without symmetry assumptions on the coefficient function.

Contribution

It establishes the existence of ground state solutions for non-autonomous fractional Choquard equations without symmetry constraints on the coefficient function.

Findings

Existence of ground state solutions under broad conditions.

No symmetry assumptions required on the coefficient function.

Applicable to a range of fractional orders and nonlinearities.

Abstract

We consider the following nonlinear fractional Choquard equation, \begin{equation}\label{e:introduction} \begin{cases} (-\Delta)^{s} u + u = (1 + a(x))(I_\alpha \ast (|u|^{p}))|u|^{p - 2}u\quad\text{ in }\mathbb{R}^N,\\ u(x)\to 0\quad\text{ as }|x|\to \infty, \end{cases} \end{equation} here $s\in (0, 1)$, $\alpha\in (0, N)$, $p\in [2, \infty)$ and $\frac{N - 2s}{N + \alpha} < \frac{1}{p} < \frac{N}{N + \alpha}$. Assume $\lim_{|x|\to\infty}a(x) = 0$ and satisfying suitable assumptions but not requiring any symmetry property on $a(x)$, we prove the existence of ground state solutions.