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Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations. (English) Zbl 1219.35292

Summary: We study existence and regularity of positive stationary solutions for a class of nonlinear pseudo-relativistic Schrödinger equations. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square-root of the Laplacian. We investigate such problems using critical point theory after transforming them to elliptic equations with nonlinear Neumann boundary conditions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35B65 Smoothness and regularity of solutions to PDEs
35B09 Positive solutions to PDEs
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