Nehari Manifold for fractional Kirchhoff system with critical nonlinearity

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)&=λf(x)|u|^{q-2}u+ \frac{2α}{α+β}\left|u\right|^{α-2}u|v|^β& \text{in } Ω,\\ \mc L_M(v)&=μg(x)|v|^{q-2}v+ \frac{2β}{α+β}\left|u\right|^α|v|^{β-2}v & \text{in } Ω,\\ u&=v=0 &\mbox{in } \mathbb{R}^{N}\setminus Ω, \end{array} \right. \end{equation} where $\mc L_M(u)=M\left(\displaystyle \int_Ω|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+bt$ with $a,b>0$ and fractional Laplacian $(-\Delta {)}^s,s\in (0,1)$. We consider that $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, {$2s<N\le 4s$} with smooth boundary, $f,g$ are sign changing continuous functions, $\lambda ,\mu >0$ are {real} parameters, $1<q<2$, $\alpha ,\beta \ge 2$ {and} $\alpha +\beta =2_s^{\ast }=2N/(N-2s)$ {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 ${\mathbb{R}}_{+}^2$.