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：本文中，据我们所知，首次研究了具有参数系数和某些幂次非线性的参数半线性椭圆特征值问题。参数系数假定与可数多个参数具有仿射依赖关系，并采用适当类别的函数序列。本文得到了基础特征对的混合导数上界估计，该估计与近期线性特征值问题所获形式相同。此估计的三个最关键要素是：基础特征对的参数解析性、基础特征对的一致有界性以及线性算子基础特征值之间的一致正差异。这三项要素均需运用新技术方法，并对非线性特征值问题进行细致研究，这些内容将在本文中呈现。作为应用，将每个参数视为均匀分布随机变量，我们采用随机平移拟蒙特卡罗格点规则估计特征对的期望值，并给出了维数无关误差界。