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：本文首次研究了带有参数系数和某些幂型非线性的半线性椭圆特征值问题。假设参数系数仿射依赖于可数多个参数，并具有适当的函数序列类。我们得到了基本特征对混合导数的上界估计，该估计形式与近期线性特征值问题得到的结果一致。该估计的三个关键要素为：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。这三个要素均需新技术，并需对非线性特征值问题进行细致研究，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，我们利用随机位移拟蒙特卡罗格点规则估计特征对的期望，并证明了维数无关的误差界。