On a fractional system of NLS-KDV equations with Hardy potentials

Abstract

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger-Korteweg-De Vries (NLS-KdV, in short) equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1} &= 2νh(x) u^{}v^{} & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} &= νh(x) u^{2} & \quad \mbox{in} ~ \mathbb{R}^{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\}, \end{aligned} \right. \end{equation*} where $s_1,s_2\in (0,1)\text{and}{\lambda }_i\in (0,{\Lambda }_{N,s_i})$ with ${\Lambda }_{N,s_i}=2{\pi }^{N/2}\frac{{\Gamma }^2(\frac{N+2s_i}{4})\Gamma (\frac{N+2s_i}{2})}{{\Gamma }^2(\frac{N-2s_i}{4})|\Gamma (-s_i)|},(i=1,2)$. By imposing certain assumptions on the parameter $\nu$ and on the function $h$, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

arXiv admin note: substantial text overlap with arXiv:2210.08260