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We study a class of nonlinear eigenvalue problems of Schr\"{o}dinger type,where the potential is singular on a set of points. Such problems are widelypresent in physics and chemistry, and their analysis is of both theoretical andpractical interest. In particular, we study the regularity of theeigenfunctions of the operators considered, and we propose and analyze theapproximation of the solution via an isotropically refined $hp$ discontinuousGalerkin (dG) method.  We show that, for weighted analytic potentials and for up-to-quarticpolynomial nonlinearities, the eigenfunctions belong to analytic-type nonhomogeneous weighted Sobolev spaces. We also prove quasi optimal a prioriestimates on the error of the dG finite element method; when using anisotropically refined $hp$ space the numerical solution is shown to convergewith exponential rate towards the exact eigenfunction. We conclude with aseries of numerical tests to validate the theoretical results.

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schr\"odinger eigenproblems