Critical growth fractional Kirchhoff elliptic problems

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-\Delta)^s u = \ds\lambda f(x)|u|^{q-2}u + \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u \;\text{in} \; \Omega, u > 0\quad \text{in} \; \Omega, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{array} \end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\te-1}$, where $a, b > 0$ are given constants. $(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{\mu,s}$ for $2< q < 2^*_{s}$. Here $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.Comment: 30 page