On the existence of a countable set of solutions in a problem of polaron theory

An investigation is made of a certain nonlinear second-order integro-differential equation having numerous applications in various areas of physics (polaron theory, the theory of many-particle quantum systems, and so on). Under certain assumptions it is proved that there is a positive solution, and, moreover, an infinite set of distinct solutions. Use is made of the Lyusternik–Shnirel'man theory of critical points and the fibering method of S. I. Pokhozhaev.