Nonlinear and Nonlocal Eigenvalue Problems: variational existence, decay properties, approximation, and universal scaling limits

We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of solutions with positive eigenvalues and unimodal eigenfunctions. We also discuss the decay properties and the numerical computations of those eigenfunctions, and conclude with two asymptotic results concerning small and large eigenvalues.