Bound and ground states of coupled "NLS-KDV" equations with Hardy potential and critical power

TL;DR

This paper investigates the existence of bound and ground states for coupled nonlinear elliptic systems with critical nonlinearities and Hardy potentials, relevant to fluid mechanics, using variational and min-max methods.

Contribution

It introduces new results on the existence of bound and ground states for coupled NLS-KDV equations with Hardy potentials, covering various coupling parameter ranges.

Findings

Ground states exist for certain coupling parameters.

Bound states are obtained via min-max methods.

Results apply to equations with critical power nonlinearities.

Abstract

We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schr\"odinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $\nu$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.