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

Semilinear elliptic eigenvalue problem: Parametric analyticity and the uncertainty quantification