# On doubly nonlocal $p$-fractional coupled elliptic system

## TL;DR

This paper investigates a nonlinear coupled system involving doubly nonlocal p-fractional Laplacians with perturbations, establishing the existence of multiple solutions using variational methods.

## Contribution

It introduces new results on the existence of multiple solutions for a doubly nonlocal p-fractional coupled elliptic system with perturbations.

## Key findings

- Proves the existence of at least two nontrivial solutions.
- Employs Nehari manifold and minimax methods for analysis.
- Addresses a complex system with nonlocal operators and perturbations.

## Abstract

\noi We study the following nonlinear system with perturbations involving p-fractional Laplacian \begin{equation*} (P)\left\{ \begin{split} (-\De)^s_p u+ a_1(x)u|u|^{p-2} &= \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u+ \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x)\; \text{in}\; \mb R^n,\\ (-\De)^s_p v+ a_2(x)v|v|^{p-2} &= \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v+ \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)\; \text{in}\; \mb R^n, \end{split} \right. \end{equation*} where $n>sp$, $0<s<1$, $p\geq2$, $\mu \in (0,n)$, $\frac{p}{2}\left( 2-\frac{\mu}{n}\right) < q <\frac{p^*_s}{2}\left( 2-\frac{\mu}{n}\right)$, $\alpha,\beta,\gamma >0$, $0< a_i \in C^1(\mb R^n, \mb R)$, $i=1,2$ and $f_1,f_2: \mb R^n \to \mb R$ are perturbations. We show existence of atleast two nontrivial solutions for $(P)$ using Nehari manifold and minimax methods.