Non variational type critical growth nonlocal system
Ashutosh Dixit, Hichem Hajaiej, Tuhina Mukherjee
TL;DR
This work studies a non-variational, multi-component system of fractional Laplacians with critical nonlinearities on $ abla extbf{R}^N$, seeking positive solutions and their multiplicities. The authors reduce the problem to an algebraic system governing synchronization coefficients and apply topological degree techniques to obtain existence and precise counts, even when a variational structure is absent. They obtain explicit multiplicity results across regimes: for $p_{ij}<2$ with small couplings there are at least $2^{n}-1$ synchronized positive solutions; for $p_{ij}=2$ a threshold-driven dichotomy yields unique, none, or continuum of synchronized solutions depending on $oldsymbol{ u}$ and $oldsymbol{eta}$; and for $2<p_{ij}<2^{*}_{s}$ there is at least one synchronized solution with detailed counts in symmetric cases that exhibit bifurcation as $p$ varies. Overall, the paper extends non-variational, topological methods to critical-growth, multi-component fractional systems and provides a nuanced, parameter-dependent map of solution multiplicities and uniqueness properties.
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 $N>2s=\max\{2s_i\}$, $s_i\in(0,1)$, $n\geq 2$, $λ_{i} \geq 0$, $α_{ij}>0$, $p_{ij}<2^{*}_{s}$, and $p_{ij}+q_{ij}=2^{*}_{s}=\min\{{\frac{2N}{N-2s_i}\}}$ for $i\neq j \in \{1,2,...,n\}$. $2^{*}_s$ called the fractional critical sobolev exponent and $2^{*}_s=2 N /(N-2s)$ for $N > 2s$ and $2^{*}_s=+\infty$ 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 $2<p_{ij}<2^{*}_{s}$.