This paper is devoted to investigating the Freidlin–Wentzell’s large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

本文致力于研究一类受小乘性噪声扰动的 McKean-Vlasov 拟线性 SPDE 的 Freidlin-Wentzell 大偏差原理。我们采用变分框架和修改后的弱收敛准则来证明 McKean-Vlasov 型 SPDE 的拉普拉斯原理，该原理等效于大偏差原理。此外，我们不假设嵌入 Gelfand 三元组的任何紧凑性条件来处理应用程序中的有界和无界域的情况。主要结果可以应用于各种 McKean-Vlasov 类型的 SPDE，例如分布相关的随机多孔介质类型方程和随机 p-拉普拉斯类型方程。