# Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

## TL;DR

This paper proves the existence and multiplicity of positive solutions for coupled fractional Schr"odinger systems with small parameters, using variational methods and critical point theory, extending known results from single equations.

## Contribution

It introduces new multiplicity results for coupled fractional Schr"odinger systems with shared potential minima, employing energy estimates, Nehari manifold, and Lusternik-Schnirelmann theory.

## Key findings

- Two positive solutions exist for the coupled system.
- Solutions concentrate at the common minimum point of the potentials.
- Conditions for existence and nonexistence of least energy solutions are identified.

## Abstract

It is well known that a single nonlinear fractional Schr\"odinger equation with a potential $V(x)$ and a small parameter $\varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schr\"odinger systems with a small parameter $\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.