On solitary wave solutions to dispersive equations with double power nonlinearities

TL;DR

This paper investigates the existence of solitary wave solutions to dispersive equations with fractional Laplacian and double power nonlinearities, extending previous work on single power cases.

Contribution

It introduces the analysis of ground states for fractional elliptic equations with double power nonlinearities, a novel extension of prior single power studies.

Findings

Existence of ground state solutions established.

Conditions for solutions depend on the nonlinear powers.

Extends known results to more complex nonlinearities.

Abstract

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