We study the existence of ground state solutions for a class of nonlinear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square root of the Laplacian. We investigate this problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.
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Vittorio Coti Zelati, Margherita Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1421–1436DOI 10.4171/RMI/763