Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Abstract

In this paper we study the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-\Delta)^s u = \frac{\lambda}{u^\gamma} + \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u \;\text{in} \; \Omega, %\quad \quad u > 0\quad \text{in} \; \Omega, \quad \quad 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 $s \in (0,1)$. $M$ models Kirchhoff-type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. $(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a real parameter, $\gamma \in (0,1)$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We prove that each positive weak solution is bounded and satisfy H\"older regularity of order $s$. Furthermore, using the variational methods and truncation arguments we prove the existence of two positive solutions.