In this paper we study the existence, multiplicity, and regularity of positive weak solutions for the following Kirchhoff–Choquard problem:

$$\begin{array}{c}M\left(\underset{{ℝ}^{2N}}{\iint }\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}dxdy\right){(-\mathrm{\Delta })}^{s}u=\frac{\lambda }{u^{\gamma }}+\left(\underset{\mathrm{\Omega }}{\int }\frac{|u(y){|}^{2_{\mu ,s}^{\ast }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{2_{\mu ,s}^{\ast }-2}u\text{in}\mathrm{\Omega },\\ u=0\text{in}{ℝ}^N\backslash \mathrm{\Omega },\end{array}$$

where Ω is open bounded domain of ${ℝ}^N$ with C² boundary, N > 2s and s ∈ (0, 1). M models Kirchhoff-type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. (− Δ)^s is fractional Laplace operator, λ > 0 is a real parameter, γ ∈ (0, 1) and $2_{\mu ,s}^{\ast }=\frac{2N-\mu }{N-2s}$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order s. Furthermore, using the variational methods and truncation arguments, we prove the existence of two positive solutions.

中文翻译：

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具有奇异和 Choquard 非线性的退化 Kirchhoff 方程的多重正解

在本文中，我们研究以下 Kirchhoff-Choquard 问题的正弱解的存在性、多重性和规律性：

$$\begin{array}{c}米\left(\underset{{ℝ}^{2N}}{\iint }\frac{|你(X)-你(是){|}^2}{|X-是{|}^{N+2秒}}dXd是\right){(-\mathrm{\Delta })}^{秒}你=\frac{\lambda }{{你}^{\gamma }}+\left(\underset{\mathrm{\Omega }}{\int }\frac{|你(是){|}^{2_{\mu ,秒}^{＊}}}{|X-是{|}^{\mu }}d是\right)|你{|}^{2_{\mu ,秒}^{＊}-2}你\text{在}\mathrm{\Omega },\\ 你=0\text{在}{ℝ}^N\backslash \mathrm{\Omega },\end{array}$$

其中Ω是开有界域 ${ℝ}^N$具有C ²边界，N  > 2 s且s  ∈ (0, 1)。M模型特别是基尔霍夫型系数，即基尔霍夫系数 M 在零时为零的退化情况。(− Δ) ^s是分数拉普拉斯算子，λ  > 0 是实参数，γ  ∈ (0, 1) 和$2_{\mu ,秒}^{＊}=\frac{2N-\mu }{N-2秒}$是 Hardy-Littlewood-Sobolev 不等式意义上的临界指数。我们证明每个正弱解都是有界的，并且满足s阶的Hölder 正则性。此外，使用变分方法和截断参数，我们证明了两个正解的存在。