Nonlinear Schrödinger Equations with Symmetric Multi-polar Potentials

Schrödinger equations with Hardy-type singular potentials have been the object of a quite large interest in the recent literature, see e.g. [1, 7, 8, 12, 15, 18, 20, 25, 26, 28]. The singularity of inverse square potentials V (x) ∼ λ|x|−2 is critical both from the mathematical and the physical point of view. As it does not belong to the Kato’s class, it cannot be regarded as a lower order perturbation of the laplacian but strongly influences the properties of the associated Schrödinger operator. Moreover, from the point of view of nonrelativistic quantum mechanics, among potentials of type V (x) ∼ λ|x|−α, the inverse square case represent a transition threshold: for λ < 0 and α > 2 (attractively singular potential), the energy is not lower-bounded and a particle near the origin in the presence a potential of this type “falls” to the center, whereas if α < 2 the discrete spectrum has a lower bound (see [21]). Moreover inverse square singular potentials arise in many fields, such as quantum mechanics, nuclear physics, molecular physics, and quantum cosmology; we refer to [17] for further discussion and motivation. The case of multi-polar Hardy-type potentials was considered in [14, 11]. In particular in [14] the authors studied the ground states of the following class of nonlinear elliptic equations with a critical power-nonlinearity and a potential exhibiting multiple inverse square singularities: