A uniqueness result for a simple superlinear eigenvalue problem

We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein-Rutmann arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.