Critical growth fractional Kirchhoff elliptic problems

Abstract

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) (-Δ)^s u = \dsλf(x)|u|^{q-2}u + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, u > 0\quad \text{in} \; Ω, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \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 $q\in (1,2_s^{\ast })$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{μ,s}$ for $2<q<2_s^{\ast }$. Here $2_s^{\ast }=\frac{2N}{N-2s}$ and $2_{\mu ,s}^{\ast }=\frac{2N-\mu }{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

30 pages