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（据我们所知）研究了具有参数系数和若干幂型非线性的参数半线性椭圆特征值问题。参数系数假设与可数多个参数存在仿射依赖关系，且函数序列属于适当类别。本文获得了基特征对混合导数的上界估计，其形式与近期线性特征值问题研究结果一致。该估计的三个核心要素为：基特征对的参数解析性、基特征对的一致有界性以及线性算子基特征值之间的一致正差异。这三个要素均需借助非线性特征值问题的新技术并经过细致研究，本文将对这些问题进行阐述。作为应用，将每个参数视为均匀分布随机变量后，我们利用随机移位的拟蒙特卡洛格点规则估计特征对的期望值，并证明了维数无关的误差界。