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）系统研究了参数化半线性椭圆特征值问题，其中参数化系数具有特定函数序列类的仿射依赖性，并包含幂次型非线性项。我们获得了基态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中得到的结论一致。该估计的三大核心要素为：基态特征对的参数解析性、基态特征对的一致有界性，以及线性算子基态特征值间的一致正差。这三方面均需发展新方法并对非线性特征值问题进行深入探讨，这正是本文的重点。作为应用实例，在将各参数视为均匀分布随机变量的假设下，我们采用随机平移拟蒙特卡洛格点法估计特征对的期望值，并证明了维度无关的误差界。