  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 widely
present in physics and chemistry, and their analysis is of both theoretical and
practical interest. In particular, we study the regularity of the
eigenfunctions of the operators considered, and we propose and analyze the
approximation of the solution via an isotropically refined $hp$ discontinuous
Galerkin (dG) method.
  We show that, for weighted analytic potentials and for up-to-quartic
polynomial nonlinearities, the eigenfunctions belong to analytic-type non
homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori
estimates on the error of the dG finite element method; when using an
isotropically refined $hp$ space the numerical solution is shown to converge
with exponential rate towards the exact eigenfunction. We conclude with a
series of numerical tests to validate the theoretical results.

Numerical Analysis;Numerical Analysis;Analysis of PDEs