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We study semilinear elliptic equations with the fractional Laplacian in$\mathbb{R}$. The equations with single power nonlinearities have been observedby Weinstein(1987), Frank--Lenzmann(2013) and so on. We focus on the equationswith double power nonlinearities and consider the existence of ground states.

On solitary wave solutions to dispersive equations with double power nonlinearities