Non variational type critical growth nonlocal system

Abstract

This study investigates the existence, uniqueness, and multiplicity of positive solutions for a system of fractional differential equations given by: \begin{equation*} (-Δ)^{s_i} u_{i}+λ_{i} u_{i}=\sum_{j=1}^{n} α_{i j}\left|u_{j}\right|^{q_{i j}}\left|u_{i}\right|^{p_{i j}-2} u_{i} , u_i\in {\mathscr{D}^{s_i,2}\left(\mathbb{R}^{N}\right)}, i=1,2,\cdots,n, \end{equation*} where $N>2s=\max \{2s_i\}$, $s_i\in (0,1)$, $n\ge 2$, $\hat{I}{»}_i\ge 0$, $\hat{I}{\pm }_{ij}>0$, $p_{ij}<2_s^{\ast }$, and $p_{ij}+q_{ij}=2_s^{\ast }=\min \{\frac{2N}{N-2s_i}\}$ for $i\ne j\in \{1,2,...,n\}$. $2_s^{\ast }$ called the fractional critical sobolev exponent and $2_s^{\ast }=2N/(N-2s)$ for $N>2s$ and $2_s^{\ast }=+\infty$ for $N=2s$ or $N<2s$. Our work establishes novel uniqueness and multiplicity results for positive solutions, applicable whether the system possesses a variational structure or not. We provide a comprehensive characterization of the exact number of positive solutions under specific parameter configurations. Our analysis shows that the positive solution set behaves differently across three distinct regimes: $p_{ij}<2$, $p_{ij}=2$, and $2<p_{ij}<2_s^{\ast }$.