Nehari Manifold for fractional Kirchhoff system with critical nonlinearity

TL;DR

This paper establishes the existence of multiple positive solutions for a fractional Kirchhoff system with critical nonlinearity using Nehari manifold techniques, addressing the challenges posed by non-local operators and critical exponents.

Contribution

It introduces a novel application of Nehari manifold methods to a fractional Kirchhoff system with critical nonlinearity, proving multiplicity results under new conditions.

Findings

Proved existence of at least two positive solutions.

Applied Nehari manifold approach to fractional Kirchhoff systems.

Addressed compactness issues with Brezis-Lieb lemma.

Abstract

In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=\lambda f(x)|u|^{q-2}u+ \frac{2\alpha}{\alpha+\beta}\left|u\right|^{\alpha-2}u|v|^\beta & \text{in } \Omega,\\ \mc L_M(v)&=\mu g(x)|v|^{q-2}v+ \frac{2\beta}{\alpha+\beta}\left|u\right|^{\alpha}|v|^{\beta-2}v & \text{in } \Omega,\\ u&=v=0 &\mbox{in } \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation} where $\mc L_M(u)=M\left(\displaystyle \int_\Omega|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+b t$ with $a, b>0$ and fractional Laplacian $(-\Delta)^{s}, s\in(0, 1)$. We consider that $\Omega$ is a bounded domain in $\mathbb{R}^N$, {$2s<N\leq 4s$} with smooth boundary, $f, g$ are sign changing continuous functions, $\lambda, \mu>0$ are {real} parameters, $1<q<2$, $\alpha, \beta\ge 2$ {and} $\alpha+\beta=2_s^*={2N}/(N-2s)$ {is a fractional critical exponent}. Using the idea of Nehari manifold technique and a compactness result based on {classical idea of Brezis-Lieb Lemma}, we prove the existence of at least two positive solutions for $(\lambda, \mu)$ lying in a suitable subset of $\mathbb R^2_+$.