In this paper, to the best of our knowledge, we make the first attempt at studying the parametric semilinear elliptic eigenvalue problems with the parametric coefficient and some power-type nonlinearities. The parametric coefficient is assumed to have an affine dependence on the countably many parameters with an appropriate class of sequences of functions. In this paper, we obtain the upper bound estimation for the mixed derivatives of the ground eigenpairs that has the same form obtained recently for the linear eigenvalue problem. The three most essential ingredients for this estimation are the parametric analyticity of the ground eigenpairs, the uniform boundedness of the ground eigenpairs, and the uniform positive differences between ground eigenvalues of linear operators. All these three ingredients need new techniques and a careful investigation of the nonlinear eigenvalue problem that will be presented in this paper. As an application, considering each parameter as a uniformly distributed random variable, we estimate the expectation of the eigenpairs using a randomly shifted quasi-Monte Carlo lattice rule and show the dimension-independent error bound.

翻译：本文首次尝试研究参数半线性椭圆特征值问题，其中包含参数系数和幂次型非线性项。参数系数假定与可数无穷多个参数具有仿射依赖关系，且参数序列函数属于适当函数类。本文得到了基本特征对的混合导数的上界估计，该估计形式与近期线性特征值问题所得结果一致。该估计的三个关键要素包括：基本特征对的参数解析性、基本特征对的一致有界性、以及线性算子基本特征值之间的一致正差。这三个要素均需采用新技术并对非线性特征值问题进行深入研究，这些内容将在本文中呈现。作为应用，本文将每个参数视为均匀分布随机变量，利用随机移位拟蒙特卡洛格子规则估计特征对的期望值，并证明了维度无关的误差界。