Convergence from power-law to logarithm-law in nonlinear fractional Schrödinger equations

In this paper, we show a connection between fractional Schrödinger equations with power-law nonlinearity and fractional Schrödinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p → 2+, converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrödinger equations.