Nonlinear Fractional Schr\"odinger Equations coupled by power-type nonlinearities

TL;DR

This paper investigates coupled nonlinear fractional Schrödinger equations, establishing existence of positive and ground state solutions under specific parameter conditions, and extends results to systems with multiple equations.

Contribution

It provides new existence results for positive and ground state solutions of coupled fractional Schrödinger systems, including multi-equation cases, under various parameter regimes.

Findings

Existence of positive radial solutions for two-equation systems.

Existence of solutions for multi-equation systems with specific parameter conditions.

Results depend on parameters like coupling constants, nonlinearity degree, and fractional order.

Abstract

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N,\\[3pt] (-\Delta)^s u_2 + \lambda_2 u_2= \mu_2 |u_2|^{2p-2}u_2+\beta |u_1|^{p}|u_2|^{p-2}u_2 \quad\text{in }\mathbb{R}^N, \end{array} \right. \end{equation*} where $ u_1,\, u_2\in W^{s,2}(\mathbb{R}^N)$, with $ N=1,\, 2,\, 3$; $\lambda_j,\,\mu_j>0$, $j=1,2$, $\beta\in \mathbb{R}$, $p\geq 2$ and $\displaystyle\frac{p-1}{2p}N<s<1$. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters $\beta, p, \lambda_j,\mu_j$, ($j=1,\, 2$) satisfy appropriate conditions. We also study the previous system with $m$-equations, $$ (-\Delta)^s u_j+ \lambda_j u_j =\mu_j |u_j|^{2p-2}u_j+ \sum_{\substack{k=1\\k\neq j}}^m\beta_{jk} |u_k|^p|u_j|^{p-2}u_j,\quad u_j\in W^{s,2}(\mathbb{R}^N);\: j=1,\ldots,m $$ where $\lambda_j,\, \mu_j>0$ for $j=1,\ldots ,m\ge 3$, the coupling parameters $\beta_{jk}=\beta_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,m$, $j\neq k$. For this system we prove similar results as for $m=2$, depending on the values of the parameters $\beta_{jk}, p, \lambda_j,\mu_j$, (for $j,k=1,\ldots,m$, $j\neq k$).