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Nonlinear Schrödinger equations with potentials vanishing at infinity. (English) Zbl 1099.35127

Summary: We deal with stationary nonlinear Schrödinger equations of the form

-ε 2 Δu+V(x)u=K(x)u p ,x N ,

where V,K>0 and p>1 is subcritical. We allow the potential V to vanish at infinity and the competing function K to be unbounded. In this framework, positive ground states may not exist. We prove the existence of at least one positive bound state solution in the semiclassical limit, i.e. for ε0. We also investigate the qualitative properties of the solution as ε0.

35Q55NLS-like (nonlinear Schrödinger) equations
35J60Nonlinear elliptic equations
81Q05Closed and approximate solutions to quantum-mechanical equations
35B25Singular perturbations (PDE)
35J20Second order elliptic equations, variational methods
47J30Variational methods (nonlinear operator equations)