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\AtBeginDocument{{\noindent\small 
Two nonlinear days in Urbino 2017,\newline 
\emph{Electronic Journal of Differential Equations},
Conference 25 (2018), pp. 39--53.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document} \setcounter{page}{39}
\title[\hfilneg EJDE-2018/conf/25\hfil
 Ground states of coupled fractional dispersive PDEs]
{Ground states of some coupled nonlocal fractional dispersive PDEs}

\author[E. Colorado \hfil EJDE-2018/conf/25\hfilneg]
{Eduardo Colorado}

\dedicatory{Dedicated to the memory of Anna Aloe}

\address{Author Eduardo Colorado \newline
Departamento de Matem\'aticas,
Universidad Carlos III de Madrid,
Avda. Universidad 30, 28911 Legan\'es, Madrid, Spain.\newline
Instituto de Ciencias Matem\'aticas,
ICMAT (CSIC-UAM-UC3M-UCM),
C/Nicol\'as Cabrera 15, 28049 Madrid, Spain}
\email{eduardo.colorado@uc3m.es, eduardo.colorado@icmat.es}

\thanks{Published September 15, 2018}
\subjclass[2010]{49J40, 35Q55, 35Q53, 35B38, 35J50} 
\keywords{Nonlinear Fractional Schr\"odinger equation; variational method;
\hfill\break\indent fractional Korteweg-de Vries equation; 
 critical point theory; ground state}

\begin{abstract}
 We show the existence of ground state solutions to the following stationary
 system coming from some coupled fractional dispersive equations such as:
 nonlinear fractional  Schr\"odinger (NLFS) equations (for dimension $n=1, 2, 3$)
 or NLFS and fractional Korteweg-de Vries equations  (for $n=1$),
 \begin{gather*}
 (-\Delta)^{s} u+ \lambda_1 u =  u^{3}+\beta uv,\quad u\in  W^{s,2}(\mathbb{R}^n)\\
 (-\Delta)^{s} v+ \lambda_2 v =  \frac 12 v^{2}+\frac 12 \beta u^2,\quad v\in  W^{s,2}(\mathbb{R}^n),
 \end{gather*}
 where $\lambda_j>0$, $j=1,2$, $\beta\in \mathbb{R}$, $n=1, 2, 3$, and
 $\frac n4< s<1$. Precisely, we  prove the existence of a positive radially
 symmetric ground state for any $\beta>0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article we  study the existence of ground state solutions to the
following stationary system coming from some coupled nonlocal fractional dispersive
equations such as: nonlinear fractional  Schr\"odinger (NLFS) equations
(for dimension $n=1,2,3$) or NLFS and fractional Korteweg-de Vries equations
(FKdV) (for $n=1$)
\begin{equation}\label{eq:Main}
\begin{gathered}
(-\Delta)^{s} u+ \lambda_1 u =  u^{3}+\beta uv,\quad 
 u\in  W^{s,2}(\mathbb{R}^n)\\
(-\Delta)^{s}  v + \lambda_2 v =  \frac 12 v^{2}+\frac 12 \beta u^2,\quad 
 v\in  W^{s,2}(\mathbb{R}^n),
\end{gathered}
\end{equation}
where $ W^{s,2}(\mathbb{R}^n)$  denotes the fractional Sobolev space,  
$n=1,2,3$. $\lambda_j>0$, $j=1,2$, the coupling factor $\beta\in \mathbb{R}$,
and the fraction $\frac n4< s<1$.

The associated critical Sobolev exponent is defined by 
$2^*_s= \frac{2n}{n-2s}$ if
 $n>2s$, and $2^{*}_s=\infty $ if $n\le 2s$. As a consequence, since 
$\frac n4<s<1$
we have that $2^*_s>4$.

It is well known that the fractional Laplacian $(-\Delta)^s$, $0<s<1$, is a
nonlocal diffusive type operator. It arises in several
 physical phenomena like flames propagation and chemical reactions in liquids,
population dynamics, geophysical fluid dynamics,
 in probability, American options in finance, in $\alpha$-stable L\'evy processes,
etc; see for instance  \cite{Applebaum,Bertoin,Cont-Tankov}.

In the one-dimensional case, when $s=1$,  \eqref{eq:Main} comes from the
following system of coupled nonlinear Sch\"odinger (NLS) and
Korteweg-de Vries (KdV) equations
\begin{equation}\label{NLS-KdV}
\begin{gathered}
if_t + f_{xx} + |f|^2f+ \beta fg  = 0\\
g_t +g_{xxx} +gg_x  + \frac 12\beta(|f|^2)_x   =  0,
\end{gathered}
\end{equation}
where $f=f(x,t)\in \mathbb{C}$ while $g=g(x,t)\in \mathbb{R}$, and
$\beta\in \mathbb{R}$ is the real coupling coefficient. System \eqref{NLS-KdV}
appears in phenomena of interactions between short and long
dispersive waves, arising in fluid mechanics, such as  the
interactions of capillary - gravity water waves. Indeed, $f$
represents the short-wave, while $g$ stands for the long-wave.
 For more details, see for instance
\cite{aa,cl,fo} and the references therein.

Looking for ``traveling-wave" solutions, namely
solutions to \eqref{NLS-KdV} of the form
\[
(f(x,t),g(x,t))=\left(e^{i\omega t}
e^{i\frac c2 x}u(x-ct),v(x-ct)\right)
\]
with $u,v$ real functions,
and choosing  $\lambda_1=\omega+\frac{c^2}{4}$,
$\lambda_2=c$, one finds that $u, v$ solve the problem
\begin{equation}\label{NLS-KdV2}
\begin{gathered}
-u'' +\lambda_1 u  =  u^3+\beta uv \\
-v'' +\lambda_2 v  =  \frac 12 v^2+\frac 12\beta u^2.
\end{gathered}
\end{equation}
This system  has been studied, among others, in \cite{aa,ab,c2,c3,dfo1,dfo2}.
Also, note that system \eqref{NLS-KdV2} corresponds to system \eqref{eq:Main}
when $s=1$ and $n=1$.

On the other hand, for $n=2, 3$, and $s=1$, system \eqref{eq:Main} corresponds to
\eqref{NLS-KdV2}
\begin{equation}\label{NLS-KdV2-n}
\begin{gathered}
-\Delta u +\lambda_1 u  =  u^3+\beta uv \\
-\Delta v +\lambda_2 v  =  \frac 12 v^2+\frac 12\beta u^2,
\end{gathered}
\end{equation}
for which the existence of bound and ground states have been studied in \cite{c2,c3}.
We observe that system \eqref{NLS-KdV2-n} can be seen as a stationary version
of a time dependent coupled NLS system
when one looks for solitary wave solutions, and  $(u,v)$ are the corresponding
standing waves solutions of \eqref{NLS-KdV2-n}
(see for instance \cite[section 6]{c3}).
It is well known that systems of
NLS-NLS time-dependent equations have applications in nonlinear Optics,
Hartree-Fock theory for Bose-Einstein
 condensates, among other physical phenomena; see for instance the earlier
mathematical works
 \cite{akanbook,ac1,ac2,acr,bt,linwei,mmp,pomp,sirakov},
the more recent list (far from complete)
 \cite{chen-zou,c1,co-oli-tav,liu-wang,soave-tavares} and references therein.
See also a close related work; \cite{c-fract}, in which was studied a close
system of coupled NLFS equations.

Here we are interested in system \eqref{eq:Main}, consisting of coupled NLS
equations involving the so called fractional Laplacian operator
(or fractional Schr\"odinger
operator, $(-\Delta)^s+\lambda \operatorname{Id}$).

Note that in dimension $n=1$, \eqref{eq:Main} can also be seen as a system of
coupled NLFS-FKdV equations.
In this  case, \eqref{eq:Main} is the corresponding stationary system when one
looks for travelling-wave solutions of
the  time-dependent system
\begin{equation}\label{fractional_NLS-KdV}
\begin{gathered}
if_t - A_s\,f + |f|^2f+ \beta fg  = 0\\
g_t -(A_s\,g)_x +gg_x  + \frac 12\beta(|f|^2)_x   =  0,
\end{gathered}
\end{equation}
where $A_s$ stands for the nonlocal fractional Laplacian $(-\Delta)^s$
in dimension $n=1$.

While for $n= 1,2,3$,  \eqref{eq:Main} can be seen as the stationary system when
one looks for standing wave solutions
of the time-dependent system of coupled NLFS equations
\begin{equation}\label{fractional_NLS-NLS}
\begin{gathered}
if_t - (-\Delta)^s f + |f|^2f+ \beta fg  = 0\\
ig_t -(-\Delta)^s g + \beta |f|^2   =  0.
\end{gathered}
\end{equation}

The main goal of this manuscript is to demonstrate that for any $\beta>0$, problem
\eqref{eq:Main} has a positive radially symmetric ground state
$\widetilde{\mathbf{u}}=(\widetilde{u},\widetilde{v})\in  W^{s,2}(\mathbb{R}^n)\times  W^{s,2}(\mathbb{R}^n)$; see
Theorems \ref{th:1} and \ref{th:ground2}.


Notice that,  for any $\beta\in \mathbb{R}$,
\eqref{eq:Main}  has a unique {\it semi-trivial} positive radially symmetric solution,
 that we denote by $\mathbf{v}_2=(0,V_2)$, where $V_2(x)$ is the unique positive
radially symmetric   ground state of
$-(\Delta)^s v+\lambda_2 v=\frac 12 v^2$
in $ W^{s,2}(\mathbb{R}^n)$; \cite{fl,fls}. Since we are interested in positive ground states,
then we have to show that they are different from the semi-trivial solution
 $\mathbf{v}_2$. To do so, we will demonstrate some properties of the semi-trivial
solution which will allow us to show that $\mathbf{v}_2$ is not a ground state.
For example, we will show that there exists a constant $\Lambda>0$ such that
   for $\beta>\Lambda$, $\mathbf{v}_2$ is a saddle point of the associated energy functional
constrained on the corresponding Nehari Manifold,
   which actually is a natural restriction. When $\beta<\Lambda$ then  $\mathbf{v}_2$
is a strict local minimum of the energy functional on the Nehari Manifold.
In this case, we exclude that $\mathbf{v}_2$ is a ground state by the construction
of a function in the Nehari Manifold with energy
lower than the energy of $\mathbf{v}_2$.
Precisely, we will demonstrate that there exists a positive radially
symmetric ground state of \eqref{eq:Main}, $\widetilde{\mathbf{u}}\neq \mathbf{v}_2$,
either: $\beta>\Lambda$ (see Theorem \ref{th:1}) or $0<\beta\le \Lambda$ and $\lambda_2$
large enough  (see Theorem \ref{th:ground2}).


This article is organized as follows.
In Section \ref{sec:not-prel} we introduce notation and preliminaries,
dealing with  some background on the fractional Laplacian and we give the
definition of ground state.  Section \ref{sec:kr}  contains some results
on the method of the {\it natural constraint} and the main properties
about the semi-trivial solution $\mathbf{v}_2$, that  we will use in the proof of
the main existence  results stated and proved in Section \ref{sec:pf2}.
 Finally, in  Section \ref{sec:final} we study the existence of ground states
for some systems with an arbitrary number of coupled equations.


\section{Preliminaries and notation}\label{sec:not-prel}

The nonlocal fractional Laplacian operator $(-\Delta)^s$ in $\mathbb{R}^n$ is
defined on the Schwartz class of functions $g\in \mathcal{S}$
through the Fourier transform,
\begin{equation}\label{fourier}
    [(-\Delta)^{\frac{\alpha}{2}}  g]^{\wedge}(\xi)=(2\pi|\xi|)^\alpha \widehat{g}(\xi),
\end{equation}
or via the Riesz potential, see for example~\cite{Landkof,Stein}.
Note that $s=1$ corresponds to the standard local
Laplacian operator. See also \cite{Laskin,dpv,fl,fls}, where the
fractional Schr\"odinger operator ($(-\Delta)^s+$Id)
is defined and are analyzed some problems dealing with.

There is another way to define this operator. If
$s=1/2$ the square root of the Laplacian acting on a function
$u$ in the whole space $\mathbb{R}^n$, can be calculated as the
normal derivative on the boundary of its harmonic extension to the
upper half-space $\mathbb{R}^{n+1}_+$, this is so-called Dirichlet
to Neumann operator. Caffarelli-Silvestre; \cite{Caffarelli-Silvestre},
  have shown that this operator can be
realized  in a local way by using one more variable and the so
called $s$-harmonic extension.

More precisely, given  $u$  a regular function defined in $\mathbb{R}^n$
we define its $s$-harmonic extension  to the upper half-space
$\mathbb{R}^{n+1}_+$ by $w={\rm Ext}_s (u)$, as the solution to the
problem
\begin{equation} \label{extension}
\begin{gathered}
-\operatorname{div}(y^{1-2s}\nabla w)=0 \quad\text{in } \mathbb{R}^{n+1}_+\\
 w=u \quad\text{on } \mathbb{R}^n\times\{y=0\}.
\end{gathered}
\end{equation}
The main relevance of the $s$-harmonic extension
comes from the following identity
 \begin{equation}
\lim_{y\to 0^+}y^{1-2s}\frac{\partial w}{\partial y}(x,y)
 =-\frac{1}{\kappa_s}(-\Delta)^{s}u(x),
\label{normalder}
\end{equation}
where $\kappa_s$ is a positive constant.
The above Dirichlet-Neumann process
\eqref{extension}-\eqref{normalder} provides a formula for the
fractional Laplacian, equivalent to that obtained from
Fourier Transform by \eqref{fourier}. In that case, the
$s$-harmonic extension  and the fractional Laplacian have
explicit expressions in terms of the Poisson and the Riesz
kernels, respectively,
\begin{equation}
  \label{poisson}
\begin{gathered}
w(x,y)=P_y^{s}*u(x)=
c_{n,s} y^{2s}\int_{\mathbb{R}^n}\frac{u(z)}{(|x-z|^{2}+y^{2})^{\frac{n+2s}{2}}}\,dz,\\
(-\Delta)^{s}u(x)= d_{n,s}\operatorname{P.V.}
\int_{\mathbb{R}^n}\frac{u(x)-u(z)}{|x-z|^{n+2s}}\,dz.
\end{gathered}
\end{equation}
The natural functional spaces are the homogeneous fractional
Sobolev space  $\dot H^{s}(\mathbb{R}^n)$ and the weighted Sobolev
space $X^{2s}(\mathbb{R}^{n+1}_+)$, that can be defined as the
completion of $\mathcal{C}_{0}^{\infty}(\overline{\mathbb{R}^{n+1}_+})$
and $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$, respectively, under the norms
\begin{gather*}
\|\phi\|^2_{X^{2s}}=\kappa_s\int_{\mathbb{R}^{n+1}_+}y^{1-2s}|\nabla
\phi(x,y)|^2\,dx dy,\\
\|\psi\|^2_{\dot{H}^{s}}=\int_{\mathbb{R}^n}|2\pi\xi|^{2s}|\widehat{\psi}(\xi)|^2\,d\xi
=\int_{\mathbb{R}^n}|(-\Delta)^{s/2}\psi(x)|^2\, dx,\\
\end{gather*}
where $\kappa_s$ is the  constant in \eqref{normalder}.
Notice that, the constants in \eqref{poisson} and $\kappa_s$
satisfy the identity $s\, c_{n,s}\kappa_{s}=d_{n,s}$, and their explicit value
can be seen  in \cite{brandle-colorado-depablo-sanchez}.

\begin{remark} \label{rmk1} \rm
The $s$-harmonic extension operator defined by \eqref{extension} is
an isometry between the spaces $\dot H^{s}(\mathbb{R}^n)$ and
$X^{2s}(\mathbb{R}^{n+1}_+)$, i.e.,
\begin{equation} \label{isometry}
\|\varphi \|_{\dot H^{s}}= \| E_s(\varphi)\|_{X^{2s}}\,,\quad \forall
 \varphi\in\dot H^{s}(\mathbb{R}^n).
\end{equation}
Even more, we have  the following inequality for
the trace $\operatorname{Tr} (w)=w(\cdot , 0)$, 
\begin{equation} \label{trace}
 \| \operatorname{Tr}(w)\|_{\dot H^{s}}\le\| w\|_{X^{2s}}\, , \quad\forall 
 w\in X^{2s}(\mathbb{R}^{n+1}_+),
\end{equation}
 see \cite{brandle-colorado-depablo-sanchez} for more details.
\end{remark}

Let us introduce
the following notation:
 \begin{itemize}
\item $E= W^{s,2}(\mathbb{R}^n)$, denotes the fractional Sobolev space, endowed with
scalar product and norm
$$
(u\mid v)_j=\int_{\mathbb{R}^n} [ (-\Delta)^{s/2} u (-\Delta)^{s/2}  v
 + \lambda_j uv] dx,\quad \|u\|_j^2=(u\mid u)_j,\quad j=1,2;
$$

\item $\mathbb{E}=E\times E$; the elements in $\mathbb{E}$ will be denoted by $\mathbf{u}
=(u,v)$; as a norm in $\mathbb{E}$ we will take
$\|\mathbf{u}\|=\|\mathbf{u}\|_{\mathbb{E}}^2=\|u\|_1^2+\|v\|_2^2$;

\item $X=X^{2s}(\mathbb{R}^{n+1}_+)$, $\mathbb{X}=X\times X$;

\item  for $\mathbf{u}\in \mathbb{E}$, the notation $\mathbf{u}\geq \mathbf{0}$, resp. $\mathbf{u}>\mathbf{0}$, means that
$u, v\geq 0$,  resp. $u,  v>0$, for all $j=1,2$.
\end{itemize}

\begin{remark} \label{rmk2} \rm
If we define
$$
\frac{\partial w}{\partial \nu^s}
= -\kappa_{s}\lim_{y\to 0^+}y^{1-2s}\,\frac{\partial w}{\partial y}\,,
$$
we can reformulate  problem \eqref{eq:Main} as
\begin{equation} \label{eq:sistema-fractional-r2+}
 \begin{gathered}
-\operatorname{div}(y^{1-2s}\nabla w_1)  =0 \quad \text{in } \mathbb{R}^{n+1}_+ \\
-\operatorname{div}(y^{1-2s}\nabla w_2)  =0 \quad \text{in } \mathbb{R}^{n+1}_+\\
\frac{\partial w_1}{\partial \nu^s} +\lambda_1 w_1
 = w_1^{3}+\beta w_1w_2\quad \text{on }\mathbb{R}\times\{y=0\} \\
 \frac{\partial w_2}{\partial \nu^s} +\lambda_2 w_2
 = \frac 12 w_2^{2}+\frac 12\beta w_1^2\quad \text{on }\mathbb{R}\times\{y=0\},
\end{gathered}
\end{equation}
with $\mathbf{w}=(w_1,w_2)\in \mathbb{X}$.

Note that if $\mathbf{w}\in \mathbb{X}$ is solution of \eqref{eq:sistema-fractional-r2+},
then $\operatorname{Tr} (\mathbf{w}(x,y))=\mathbf{w}(x,0)\in \mathbb{E}$ is a solution of \eqref{eq:Main},
or equivalently, if $\mathbf{u}\in \mathbb{E}$ is a solution of \eqref{eq:Main},
then ${\rm Ext}_s (\mathbf{u})\in \mathbb{X}$
is a solution of \eqref{eq:sistema-fractional-r2+}.

The introduction of this problem is only for the interested reader.
As we will see along the paper, it is not necessary to make use of problem
\eqref{eq:sistema-fractional-r2+}, i.e., all the results for \eqref{eq:Main}
are going to be proved without using the
$s$-harmonic extension to the upper half space, $E_s (\cdot)$.
\end{remark}

For $\mathbf{u}=(u,\, v)\in \mathbb{E}$,  we set
\begin{equation} \label{eq:II}
\begin{gathered}
I_1(u) = \frac 12 \int_{\mathbb{R}^n} (|(-\Delta)^{s/2}  u|^2+\lambda_1 u^2)dx
-\frac{1}{4}\, \int_{\mathbb{R}^n} u^4dx,\\
I_2(v)  = \frac 12 \int_{\mathbb{R}^n} (|(-\Delta)^{s/2}  v|^2+\lambda_2 v^2)dx -\frac{1}{6}
 \int_{\mathbb{R}^n} v^3dx,\\
\Phi (\mathbf{u})= I_1(u)+I_2(v)- \frac 12\beta \int_{\mathbb{R}^n} u^2v\,dx.
\end{gathered}
\end{equation}
We also write
$$
G_\beta (\mathbf{u})=\frac 14\, \int_{\mathbb{R}^n} u^4dx+\frac 16\, \int_{\mathbb{R}^n} v^3dx
+ \frac 12\beta \int_{\mathbb{R}^n} u^2v\,dx,
$$
and using this notation we can rewrite the energy functional as
$$
\Phi (\mathbf{u})=\frac 12\|\mathbf{u} \|^2 -G_\beta (\mathbf{u}),\quad \mathbf{u}\in \mathbb{E}.
$$
We observe that $G_\beta$ makes
sense because $\frac n4<s<1\Rightarrow 2^*_s>4$ which implies
the continuous Sobolev embedding $E\hookrightarrow L^{4}(\mathbb{R}^n)$.
Even more, any critical point $\mathbf{u}\in \mathbb{E}$ of $\Phi$,  gives rise to a solution of
\eqref{eq:Main}.

\begin{definition}\label{def:ac} \rm
A non-negative critical point  $\widetilde{\mathbf{u}}\in \mathbb{E}\setminus\{\mathbf{0}\}$ is called a
 ground state of \eqref{eq:Main}
 if its energy $\Phi(\widetilde{\mathbf{u}})$ is minimal among all the non-trivial critical
points of $\Phi$.
\end{definition}

\section{Nehari manifold and properties of $\mathbf{v}_2$}\label{sec:kr}
Let us set
$$
\Psi(\mathbf{u})=(\nabla\Phi(\mathbf{u})|\mathbf{u})=(I_1'(u)|u)+(I_2'(v)|v)-\frac 32\,\beta \int_{\mathbb{R}^n} u^2v\,dx.
$$
We define the Nehari manifold  by
$$
\mathcal{N} =\{ \mathbf{u}\in \mathbb{E}\setminus\{\mathbf{0}\}: \Psi (\mathbf{u})=0\}.
$$
Then, one has that
\begin{equation}\label{eq:gamma}
(\nabla \Psi(\mathbf{u}) \mid \mathbf{u})= - \|\mathbf{u} \|^2-\int_{\mathbb{R}^n} u^4\,dx<0\quad\forall  \mathbf{u}\in \mathcal{N},
\end{equation}
thus $\mathcal{N}$ is a smooth manifold locally near any point $\mathbf{u}\not= \mathbf{0}$ with
$\Psi(\mathbf{u})=0$. Moreover,
$\Phi''(\mathbf{0})= I_1''(0)+I_2''(0)$ is positive definite, so we infer that
 $\mathbf{0}$ is a strict local minimum for $\Phi$. As a consequence,
$\mathbf{0}$ is an isolated point of the set $\{\Psi(\mathbf{u})=0\}$, proving that
$\mathcal{N}$ is a smooth complete manifold of codimension $1$,
and on the other hand there exists a constant $\rho>0$ so that
\begin{equation} \label{eq:bound1}
\|\mathbf{u}\|^2>\rho\quad\forall \mathbf{u}\in \mathcal{N}.
\end{equation}
Furthermore, by \eqref{eq:gamma} and \eqref{eq:bound1} we can show that
$\mathbf{u}\in \mathbb{E}\setminus\{\mathbf{0}\}$
is a critical point of $\Phi$ if and only if $\mathbf{u}\in\mathcal{N}$ is a critical point
of $\Phi$ constrained on $\mathcal{N}$.
As a consequence, we have the following result.

 \begin{lemma}\label{pr:ac}
 $\mathbf{u} \in \mathbb{E}$ is a non-trivial critical point of $\Phi$ if and only if
 $\mathbf{u}\in \mathcal{N}$ and is a constrained critical point of $\Phi$ on $\mathcal{N}$.
 \end{lemma}

\begin{remark}\label{rem:obs1} \rm
(i) By the previous arguments, the Nehari manifold $\mathcal{N}$ is a natural constraint
of $\Phi$. Also, it is relevant to point out that working on the Nehari manifold,
the functional $\Phi$ satisfies the following expression,
\begin{equation} \label{eq:restriction0}
\Phi|_{\mathcal{N}}(\mathbf{u})= \frac 16\|\mathbf{u}\|^2+\frac{1}{12}\int_{\mathbb{R}^n} u^4dx=:F(\mathbf{u}),
\end{equation}
then using  \eqref{eq:bound1} into \eqref{eq:restriction0} we
obtain
\begin{equation}\label{eq:restriction}
\Phi(\mathbf{u})\ge   \frac 16\|\mathbf{u}\|^2>\frac 16 \rho\quad \forall  \mathbf{u}\in\mathcal{N}.
\end{equation}
Therefore, by \eqref{eq:restriction} the functional $\Phi$ is bounded from below
on $\mathcal{N}$, as a consequence we will
minimize it on the Nehari manifold.
 To do so, a remark about compactness is  in order.

(ii) Analyzing the Palais-Smale (PS) condition, we remember that
working on the radial setting, $H=E_{radial}$, the embedding of $H$ into
$L^4(\mathbb{R}^n)$ is compact for $n=2, 3$,
but in dimension $n=1$, the embedding of $E$ or $H$ into $L^q(\mathbb{R})$ for
$2<q<2^*_s$ is not compact; see \cite[Remarque I.1]{Lions-JFA82}.
However, we will analyze all the dimensional cases $n=1,2,3$, proving
that for a PS sequence of $\Phi$ on $\mathcal{N}$, we can find a subsequence for which the
weak limit is non-trivial and it is a solution of \eqref{eq:Main}.
This fact jointly with some properties of the Schwarz symmetrization will allow
us to demonstrate the existence of positive radially symmetric ground states
to \eqref{eq:Main}.
Notice that one could also try to work in the cone of non-negative radially
decreasing functions, where one has the required compactness, in the
one-dimensional case, thanks to Berestycki and Lions \cite{BL},
 but this is not our approach.
\end{remark}

\begin{remark} \label{rem:non-trivial} \rm
It is known \cite{fl,fls}  that the equation
\begin{equation} \label{eq:soliton-s}
(-\Delta)^{s} v +v= v^2,
\end{equation}
with $v\in E$, $v\not\equiv 0$, has a unique  radially symmetric and positive
solution, that we will denote by $V$. Indeed $V$ is a non-degenerate ground
state of \eqref{eq:soliton-s} in $H$.

Clearly,  for every $\beta\in \mathbb{R}$, \eqref{eq:Main}
already possesses a {\it semi-trivial} solution given by
 $$
 \mathbf{v}_2 = (0,V_2),
 $$
 where
\begin{equation} \label{reescale_v2}
V_2(x)=2\lambda_2 V(\lambda_2^{1/{2s}}\,x)
\end{equation}
is the unique positive radially symmetric solution of $(-\Delta)^s v+\lambda_2
v=\frac 12 v^2$ in $E$.
\end{remark}

To study some useful properties of $\mathbf{v}_2$, we define
  the corresponding Nehari manifold associated to $I_2$ in \eqref{eq:II},
$$
\mathcal{N}_2 =\big\{v\in E : (I_2'(v)|v)=0\big\}
=\big\{v\in E : \|v\|_2^2 -\frac 12\int_{\mathbb{R}^n} v^3dx=0\big\}.
$$
Let us denote $T_{\mathbf{v}_2}\mathcal{N}$ the tangent space  to  $\mathcal{N}$ on $\mathbf{v}_2$. Since
$$
\mathbf{h}=(h_1,h_2)\in  T_{\mathbf{v}_2}\mathcal{N} \Longleftrightarrow
(V_2|h_2)_2= \frac 34\int_{\mathbb{R}^n} V_2^2h_2\,dx ,
$$
it follows that
\begin{equation}\label{eq:tang1}
(h_1,h_2)\in T_{\mathbf{v}_2} \mathcal{N}  \Longleftrightarrow h_2\in T_{V_2} \mathcal{N}_2.
\end{equation}

\begin{proposition}\label{prop:5}
There exists $\Lambda>0$ such that:
\begin{itemize}
\item[(i)]  if $\beta< \Lambda$, then $\mathbf{v}_2$ is a strict minimum of $\Phi$
 constrained on $\mathcal{N}$,
\item[(ii)] for any   $\beta>\Lambda$, then $\mathbf{v}_2$ is a saddle point of $\Phi$
 constrained on $\mathcal{N}$ with $\inf_{\mathcal{N}}\Phi <\Phi(\mathbf{v}_2)$.
\end{itemize}
\end{proposition}

\begin{proof}
First, we observe that if $D^2\Phi_{\mathcal{N}}$ denotes the second
derivative of $\Phi$ constrained on $\mathcal{N}$. Using that $\Phi'(\mathbf{v}_2)=0$
 we have that $D^2\Phi_{\mathcal{N}}
(\mathbf{v}_2)[\mathbf{h}]^2=\Phi'' (\mathbf{v}_2)[\mathbf{h}]^2$ for all $\mathbf{h}\in
T_{\mathbf{v}_2}\mathcal{N}$.


(i) We define
\begin{equation} \label{eq:Lambda}
\Lambda=\inf_{\varphi\in E\setminus\{
0\}}\frac{\|\varphi\|_1^2}{\int_{\mathbb{R}^n} V_2\varphi^2dx}.
\end{equation}
We have that for $\mathbf{h}\in  T_{\mathbf{v}_2}\mathcal{N}$,
\begin{equation} \label{eq:Phi-segunda}
\Phi''(\mathbf{v}_2)[\mathbf{h}]^2 =\|h_1\|_1^2 +I_2''(V_2)[h_2]^2-\beta\int_{\mathbb{R}^n} V_2
h_1^2dx.
\end{equation}
Let us take $\mathbf{h}=(h_1,h_2)\in T_{\mathbf{v}_2}\mathcal{N}$, by
\eqref{eq:tang1} $h_2\in T_{V_2} \mathcal{N}_2$, then using that $V_2$ is
the minimum of $I_2$ on $\mathcal{N}_2$, there exists a  constant $c>0$ such
that
\begin{equation} \label{eq:minimo-pos}
I_2'' (V_2)[h_2]^2\ge c\|h_2\|_2^2.
\end{equation}
 From \eqref{eq:minimo-pos} jointly with
\eqref{eq:Phi-segunda}, for $\beta<\Lambda$, there exists another constant
$c_1>0$ such that, 
\begin{equation} \label{minimo} 
\Phi''(\mathbf{v}_2)[\mathbf{h}]^2 \ge c_1(\|h_1\|_1^2 +\| h_2\|^2), 
\end{equation}
which proves that  $\mathbf{v}_2$ is a  strict local minimum of $\Phi$ on $\mathcal{N}$.

(ii) According to \eqref{eq:tang1}, $\mathbf{h}=(h_1,0)\in T_{\mathbf{v}_2}\mathcal{N}$ for any
$h_1\in E$.
We have that, for $\beta>\Lambda$, there exists $\widetilde{h}\in E$ with
$$
\Lambda< \frac{\|\widetilde{h}\|_1^2}{\int_{\mathbb{R}^n} V_2\widetilde{h}^2dx}<\beta,
$$
thus, taking $\mathbf{h}_0=(\widetilde{h},0)\in T_{\mathbf{v}_2}\mathcal{N}$, by \eqref{eq:Phi-segunda} we find
\begin{equation} \label{eq:*1}
\Phi''(\mathbf{v}_2)[\mathbf{h}_0]^2 =\|\widetilde{h}\|_1^2 -\beta\int_{\mathbb{R}^n} V_2 \widetilde{h}^2dx<0.
\end{equation}
On the other hand,
by \eqref{eq:tang1}, and using again that $V_2$ is
the minimum of $I_2$ on $\mathcal{N}_2$, we have that there exists $c>0$ such that
$$
I_2''(V_2)[h]^2\ge c\|h\|_2^2, \:\forall  h\in T_{V_2}\mathcal{N}_2.
$$
Finally, by \eqref{eq:Phi-segunda}, $\Phi''(\mathbf{v}_2)[(0,h)]^2=I_2''(V_2)[h]^2$
for any $h\in T_{V_2}\mathcal{N}_2$.
Thus we have that there exists a  constant $c>0$ such that
$$
\Phi''(\mathbf{v}_2)[\mathbf{h}_1]^2\ge c\|\mathbf{h}_1\|^2,\quad \forall \mathbf{h}_1=(0,h_1)\in T_{\mathbf{v}_2}\mathcal{N}.
$$
\end{proof}

\section{Ground state solutions}\label{sec:pf2}

The first result on the existence of ground states is given for the coupling
parameter $\beta>\Lambda$ in the following theorem.

\begin{theorem}\label{th:1}
Assume  $\beta>\Lambda$, then $\Phi$  has a positive radially symmetric ground
state $\widetilde{\mathbf{u}}$, and there holds
$\Phi(\widetilde{\mathbf{u}})<\Phi(\mathbf{v}_2)$.
\end{theorem}

\begin{proof}
By the Ekeland's variational principle;
\cite{eke}, there exists a PS sequence $\{\mathbf{u}_k\}_{k\in
\mathbb{N}}\subset\mathcal{N}$, i.e.,
\begin{gather} \label{eq:PS1}
\Phi (\mathbf{u}_k)\to c_\mathcal{N}=\inf_{\mathcal{N}}\Phi, \\
\nabla_{\mathcal{N}}\Phi(\mathbf{u}_k)\to 0.
\end{gather}
By \eqref{eq:restriction0} and \eqref{eq:PS1}, we find that $\{\mathbf{u}_k\}$ is a
bounded sequence on $\mathbb{E}$, hence for a subsequence, we can assume that
\begin{gather} \label{eq:weakly}
\mathbf{u}_k\rightharpoonup \mathbf{u}_0\quad\text{weakly in } \mathbb{E}, \\
 \label{eq:local}
\mathbf{u}_k\to \mathbf{u}_0\quad \text{strongly in }\mathbb{L}^q_{\rm loc}(\mathbb{R})
=L^q_{\rm loc}(\mathbb{R})\times  L^q_{\rm loc}(\mathbb{R})\quad \forall  1\le q<2^*_s,
\end{gather}
and also $\mathbf{u}_k\to \mathbf{u}_0$ a. e. in $\mathbb{R}^n$. Since $\mathcal{N}$ is closed we have that
$\mathbf{u}_0\in\mathcal{N}$, even more, using that $\mathbf{0}$ is an isolated point
the set $\{ \Psi (\mathbf{u})=0\}$ we infer that $\mathbf{u}_0\neq \mathbf{0}$.
On the other hand,
the constrained gradient satisfies
\begin{equation} \label{eq:to0}
\nabla_{\mathcal{N}}\Phi (\mathbf{u}_k)=\Phi' (\mathbf{u}_k)-\eta_k \Psi'(\mathbf{u}_k)\to 0,
\end{equation}
where $\eta_k$ is the
corresponding Lagrange multiplier. Taking the scalar product with
$\mathbf{u}_k$ in \eqref{eq:to0}, since $\mathbf{u}_k\in\mathcal{N}$ we have that
$(\Phi'(\mathbf{u}_k)\mid \mathbf{u}_k)=\Psi(\mathbf{u}_k)=0$,
then we infer that $\eta_k (\Psi'(\mathbf{u}_k)\mid
\mathbf{u}_k)\to 0$; this jointly with
\eqref{eq:gamma},\eqref{eq:restriction0} and the fact that
$\|\Psi'(\mathbf{u}_k)\|\leq C<\infty$ imply  that $\eta_k\to 0$ and therefore
$\Phi'(\mathbf{u}_k)\to 0$.

As a consequence of the discussion above, although we do not know that
$\mathbf{u}_k\to\mathbf{u}_0$ in $\mathbb{E}$, we infer that $\mathbf{u}_0\in\mathcal{N}$
is a non-trivial critical point of $\Phi$ and by Lemma \ref{pr:ac}
it is also a non-trivial critical point of $\Phi$ on $\mathcal{N}$.

Moreover, using that  $\mathbf{u}_0\in\mathcal{N}$ jointly with \eqref{eq:restriction0}
 and the Fatou's Lemma,  we find
$$
\Phi (\mathbf{u}_0)  =   F(\mathbf{u}_0)
 \le  \liminf_{k\to\infty} F(\mathbf{u}_k)\\
 = \liminf_{k\to\infty}\Phi(\mathbf{u}_k)= c_\mathcal{N}.
$$

As a consequence, $\mathbf{u}_0$ is a least energy solution of \eqref{eq:Main}.
By Proposition \ref{prop:5}-(ii) we know that
necessarily $\Phi(\mathbf{u}_0)<\Phi(\mathbf{v}_2)$. Additionally, by the maximum
principle in the fractional setting; \cite{cabre-sire}, applied to
the second equation in \eqref{eq:Main}, we have that
$v_0>0$.
To show that also  $u_0>0$,
first we prove the following result.
\smallskip

\noindent{\it Claim.} We can assume without loss of generality that ${u}_0\ge 0$.

To prove this, we consider
$|\mathbf{u}_0|=(|{u}_0|,v_0)$, then we have two cases:
\smallskip

\noindent\textbf{Case 1.}
If $|\mathbf{u}_0|\in\mathcal{N}$, by the Stroock-Varopoulos inequality
\cite{stroock, varop},
\begin{equation} \label{eq:stroock-var}
\|(-\Delta)^{s/2} (|u|)\|_{L^2}\le
\|(-\Delta)^{s/2} (u)\|_{L^2},
\end{equation}
we have, in particular, that
 $\||u|\|_1\le \| u\|_1$, then we obtain
$$
\Phi (|\mathbf{u}_0|)\le \Phi(\mathbf{u}_0)=c_\mathcal{N}.
$$
Then, by similar arguments as in \cite[Theorem 4.3]{Willem},
we have that $|\mathbf{u}_0|$ is a non-negative ground state.
\smallskip

\noindent\textbf{Case 2.} If $|\mathbf{u}_0|\not\in\mathcal{N}$, we take the unique $t>0$,
$t\neq 1$ such that $t|\mathbf{u}_0|\in\mathcal{N}$, which comes from
\begin{equation} \label{eq:tpos}
\|\, |\mathbf{u}_0|\, \|^2=t^2\int_{\mathbb{R}^n} u_0^4dx+t\Big(
\frac 12\int_{\mathbb{R}^n} v_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n} {u}_0^2{v}_0 \,dx\Big).
\end{equation}
Since $\mathbf{u}_0\in\mathcal{N}$, then
\begin{equation} \label{eq:tpos2}
\|\, \mathbf{u}_0\, \|^2=\int_{\mathbb{R}^n} {u}_0^4dx+
\frac 12\int_{\mathbb{R}^n} {v}_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n}
{u}_0^2{v}_0 \,dx.
\end{equation}
By \eqref{eq:tpos}, \eqref{eq:tpos2} and again the Stroock-Varopoulos
inequality \eqref{eq:stroock-var}, we infer that
\begin{equation} \label{eq:tpos3}
\begin{aligned}
 & t^2\int_{\mathbb{R}^n} {u}_0^4dx+t\Big(
\frac 12\int_{\mathbb{R}^n} {v}_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n} {u}_0^2{v_0} \,dx\Big)\\
& \le \int_{\mathbb{R}^n} {u}_0^4dx+ \frac 12\int_{\mathbb{R}^n} {v}_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n}
{u}_0^2{v}_0 \,dx.
\end{aligned}
\end{equation}
Using that $t\neq 1$, as a consequence of \eqref{eq:tpos3} we deduce that
$0<t<1$ and the inequality in \eqref{eq:tpos3} is strict.
Hence, by \eqref{eq:restriction0} jointly with \eqref{eq:stroock-var}
and $t<1$  we obtain
\begin{align*}
\Phi (t|\mathbf{u}_0|) & =  t^2\|\, |\mathbf{u}_0|\, \|^2 +t^4\frac{1}{12}\int_{\mathbb{R}^n} u_0^4dx\\
& <   \|\, |\mathbf{u}_0|\, \|^2 +\frac{1}{12}\int_{\mathbb{R}^n} u_0^4dx\\
& \le  \Phi(\mathbf{u}_0)=c_\mathcal{N}.
\end{align*}
This is a contradiction because  $t|\mathbf{u}_0|\in\mathcal{N}$. Therefore
 $|\mathbf{u}_0|\in\mathcal{N}$ and the claim is proved.
\smallskip

Once we can assume without loss of generality that ${u}_0\ge 0$, by the maximum
principle applied to the first equation in \eqref{eq:Main}
we find ${u}_0> 0$ proving that indeed $\mathbf{u}_0$ is a positive ground state.

To complete the proof, we have to show that the ground state is indeed
radially symmetric.

If $\mathbf{u}_0$ is not radially symmetric, we set
$\widetilde{\mathbf{u}}=\mathbf{u}_0^\star=(u_0^\star,v_0^\star)$,
where $u_0^\star,\,{v}_0^\star$ denote the Schwarz symmetric functions
associated to $u_0, v_0$ respectively. By the properties of the
Schwarz symmetrization; see for instance \cite{fmm} for the fractional setting and
\cite{ban} for the classical one, there hold
\begin{equation} \label{eq:primera}
\|{\mathbf{u}}^\star\|^2\le \|{\mathbf{u}}\|^2, \quad
G_\beta({\mathbf{u}}^\star)\ge G_\beta({\mathbf{u}}).
\end{equation}
Furthermore,  there exists a unique $t_\star>0$ such that
$t_\star\,\widetilde{\mathbf{u}}\in {\mathcal{N}}$. If $t_\star=1$, by \eqref{eq:primera} we have
$\Phi (\widetilde{\mathbf{u}})\le \Phi (\mathbf{u}_0)=c_\mathcal{N}$ with $\widetilde{\mathbf{u}}\in\mathcal{N}$
thus $\widetilde{\mathbf{u}}$ is a positive radially symmetric ground state of \eqref{eq:Main}.

On the contrary, i.e., if $t_\star\neq 1$, as in \eqref{eq:tpos}, $t_\star$
comes from
\begin{equation} \label{eq:t}
\| \widetilde{\mathbf{u}}\|^2=
t_\star^2\int_{\mathbb{R}^n} (u_0^\star)^4dx+t_\star\Big( \frac 12\int_{\mathbb{R}^n}
(v_0^\star)^3dx+\frac 32\beta \int_{\mathbb{R}^n} (u_0^\star)^2v_0^\star
\,dx\Big).
\end{equation}
Because $\mathbf{u}_0\in\mathcal{N}$,
\eqref{eq:primera}, \eqref{eq:t},  that $\mathbf{u}_0>\mathbf{0}$ and that
$t_\star>0$, we find
\begin{equation} \label{t_star}
\begin{aligned}
&   \int_{\mathbb{R}^n} u_0^4\,dx+ \frac 12\int_{\mathbb{R}^n} v_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n} u_0^2v_0\,dx \\
& \ge   t_\star^2\int_{\mathbb{R}^n} u_0^4dx+t_\star\Big( \frac 12\int_{\mathbb{R}^n}
v_0^3\,dx+\frac 32\beta \int_{\mathbb{R}^n} u_0^2v_0 \,dx\Big).
\end{aligned}
\end{equation}
Thus, using that $0<t_\star\neq 1$ in \eqref{t_star}, we obtain  $0<t_\star< 1$,
this and \eqref{eq:primera} show that
\begin{equation} \label{eq:previa}
\Phi(t_\star\,\widetilde{\mathbf{u}})= \frac 16
t_\star^2\|{\mathbf{u}}^\star\|^2+\frac{1}{12}t_\star^4\int_{\mathbb{R}^n} (u_0^\star)^4\,dx <
\frac 16 \|\mathbf{u}_0\|^2+\frac{1}{12}\int_{\mathbb{R}^n} u_0^4\,dx=\Phi(\mathbf{u}_0)=c_\mathcal{N},
\end{equation}
with $t_\star\widetilde{\mathbf{u}}\in\mathcal{N}$ which is a contradiction with \eqref{eq:previa},
proving that $t_\star=1$ and as above, the proof is complete.
\end{proof}

The second result about existence of ground states cover the range $0<\beta\le\Lambda$,
 provided  $\lambda_2$ is large enough.

\begin{theorem}\label{th:ground2}
There exists $\Lambda_2>0$ such that if $\lambda_2>\Lambda_2$, System \eqref{eq:Main} has a radially symmetric ground state
$\widetilde{\mathbf{u}}>\mathbf{0}$ for every $0<\beta\le\Lambda$.
\end{theorem}

\begin{proof}
Arguing  as in the proof of Theorem \ref{th:1},
we prove that there exists a radially symmetric ground state
$\widetilde{\mathbf{u}}\ge \mathbf{0}$. Moreover, in Theorem \ref{th:1} for $\beta>\Lambda$ we
proved that $\widetilde{\mathbf{u}}>\mathbf{0}$. Now we need to show that for $0<\beta\le \Lambda$
indeed $\widetilde{\mathbf{u}}>\mathbf{0}$ which follows by the maximum principle
provided $\widetilde{\mathbf{u}}\neq \mathbf{v}_2$. Taking into account Proposition
\ref{prop:5}-(i), $\mathbf{v}_2$ is a strict local minimum of $\Phi$ on $\mathcal{N}$,
and this does not guarantee
that $\mathbf{u}_0\not \equiv\mathbf{v}_2$. Following \cite{c3}, the  idea
consists on the construction of a function
$\mathbf{u}_0=(u_0,v_0)\in\mathcal{N}$ with $\Phi(\mathbf{u}_0)<\Phi(\mathbf{v}_2)$. To do so,
since $\mathbf{v}_2=(0,V_2)$ is a local minimum of $\Phi $ on $\mathcal{N}$
provided $0<\beta<\Lambda$, we cannot find $\mathbf{u}_0$ in a neighborhood of
$\mathbf{v}_2$ on $\mathcal{N}$. Thus, we define $\mathbf{u}_0=t(V_2,V_2)$ where  $t>0$ is
the unique value such that $\mathbf{u}_0\in \mathcal{N}$.

Now, we  show that
$$
\mathbf{u}_0=t(V_2,V_2)\in\mathcal{N}\quad \text{with } \Phi(\mathbf{u}_0)<\Phi(\mathbf{v}_2),
$$
for $\lambda_2$ large enough.

Notice that $t>0$ comes from $\Psi (\mathbf{u}_0)=0$, i.e.,
\begin{equation} \label{eq:condicion1}
t^2\|(V_2,V_2)\|^2-t^4\int_{\mathbb{R}^n} V_2^4\,dx
-\frac{1}{2}t^3(1+3\beta)\int_{\mathbb{R}^n} V_2^3\,dx=0.
\end{equation}
We also have
\begin{equation} \label{eq:norma doble}
\|(V_2,V_2)\|^2=2\|V_2\|_2^2+(\lambda_1-\lambda_2)\int_{\mathbb{R}^n} V_2^2\,dx.
\end{equation}
Moreover, since $V_2\in\mathcal{N}_2$, we have
\begin{equation} \label{eq:norma simple}
\|V_2\|_2^2-\frac{1}{2}\int_{\mathbb{R}^n} V_2^3\,dx=0.
\end{equation}
Substituting \eqref{eq:norma doble} and \eqref{eq:norma simple} in
\eqref{eq:condicion1} it follows
\begin{equation} \label{eq:condicion2}
\begin{aligned}
& t^2\Big(\int_{\mathbb{R}^n} V_2^3\,dx+(\lambda_1-\lambda_2) \int_{\mathbb{R}^n}
V_2^2\,dx\Big)
&-t^4\int_{\mathbb{R}^n} V_2^4\,dx -\frac{1}{2}t^3(1+3\beta)\int_{\mathbb{R}^n} V_2^3\,dx=0.
\end{aligned}
\end{equation}
Hence, applying the scaling \eqref{reescale_v2}  yields
\begin{equation} \label{cambio p}
\int_{\mathbb{R}^n} V_2^r\,dx=2^r\lambda_2^{r-\frac{n}{2s}}\int_{\mathbb{R}^n} V^r\,dx.
\end{equation}
 Subsequently, substituting \eqref{cambio p} for $r=2,3,4$ into
\eqref{eq:condicion2} and dividing in \eqref{eq:condicion2} by
$2^3\lambda_2^{3-\frac{n}{2s}}t^2$
we have
\begin{equation} \label{eq:condicion}
\int_{\mathbb{R}^n} V^3\,dx+\frac{\lambda_1-\lambda_2}{2\lambda_2}\int_{\mathbb{R}^n}
V^2\,dx-2\lambda_2t^2\int_{\mathbb{R}^n} V^4\,dx-\frac{1}{2}t(1+3\beta)\int_{\mathbb{R}^n}
V^3\,dx=0.
\end{equation}
Moreover, by \eqref{eq:restriction0},
\eqref{eq:norma doble} and \eqref{eq:norma simple} we find
respectively the expressions
\begin{gather} \label{eq: forma de Phi(w)}
\Phi(\mathbf{u}_0)=\frac{1}{6}t^2\Big( \int_{\mathbb{R}^n}
V_2^3\,dx+(\lambda_1-\lambda_2) \int_{\mathbb{R}^n} V_2^2\,dx\Big)
+\frac{1}{12}t^4\int_{\mathbb{R}^n} V_2^4\,dx, \\
\label{eq: forma de Phi(bv_2)}
\Phi(\mathbf{v}_2)=I_2(V_2)=\frac{1}{2}\|V_2\|_2^2-\frac{1}{6}\int_{\mathbb{R}^n}
V_2^3=\frac{1}{12}\int_{\mathbb{R}^n} V_2^3.
\end{gather}
By \eqref{eq: forma de Phi(w)}, \eqref{eq: forma de Phi(bv_2)} we have
$\Phi(\mathbf{u}_0)< \Phi(\mathbf{v}_2)$ is equivalent to
\begin{equation} \label{40}
\begin{aligned}
&\frac{1}{6}t^2\Big( \int_{\mathbb{R}^n}
V_2^3\,dx+(\lambda_1-\lambda_2) \int_{\mathbb{R}^n} V_2^2\,dx\Big) \\
&+\frac{1}{12}t^4\int_{\mathbb{R}^n} V_2^4\,dx- \frac{1}{12}\int_{\mathbb{R}^n} V_2^3\,dx <0,
\end{aligned}
\end{equation}
and then, applying again \eqref{cambio p} and multiplying
\eqref{40} by $6\lambda_2^{\frac{n}{2s}-3}$, we actually have
\begin{equation} \label{desigualdad de w}
t^2\Big( \int_{\mathbb{R}^n} V^3\,dx+\frac{\lambda_1-\lambda_2}{\lambda_2}\int_{\mathbb{R}^n} V^2\,dx\Big)
+\frac{1}{2}t^4\lambda_2\int_{\mathbb{R}^n} V^4\,dx-\frac{1}{2}\int_{\mathbb{R}^n} V^3\,dx<0.
\end{equation}
For $\lambda_2$ large enough we find that \eqref{eq:condicion} will provide
us with \eqref{desigualdad de w}.
Therefore, there exists a positive constant $\Lambda_2$ such that for $\lambda_2>\Lambda_2$
inequality \eqref{desigualdad de w} holds,
proving that
$$
\Phi(\widetilde{\mathbf{u}})\leq\Phi(\mathbf{u}_0)< \Phi(\mathbf{v}_2).
$$
Finally, this  shows that $\widetilde{\mathbf{u}}\neq \mathbf{v}_2$ and we finish.
\end{proof}

\section{Systems with more than $2$ equations}\label{sec:final}

In this last subsection, we deal with some extended systems of \eqref{eq:Main}
to more than two equations.
We start with the study of the
following system coming from NLFS-2FKdV equations if $n=1$ or 3NLFS equations
if $n=1,2,3$,
\begin{equation}\label{NLS-KdV2-3}
\begin{gathered}
(-\Delta)^s u +\lambda_0 u  =  u^3+\beta_1 uv_1+\beta_2uv_2,\\
(-\Delta)^s v_1 +\lambda_1 v_1  =  \frac 12 v_1^2+\frac 12\beta_1 u^2,\\
(-\Delta)^s v_2+\lambda_2 v_2  =  \frac 12v_2^2+\frac 12\beta_2 u^2,
\end{gathered}
\end{equation}
where $u,v_1,v_2\in E$.
This system can be seen as a perturbation of \eqref{eq:Main} when
$|\beta_1|$ or $ |\beta_2|$ is small.

We use similar notation as in previous sections  with natural
meaning, for example, $\mathbb{E}= E\times E \times E$,
$\mathbf{0}=(0,0,0)$,
\begin{gather} \label{eq:Phi3}
\Phi (\mathbf{u})=\frac 12 \|\mathbf{u}\|^2-\frac 14\int_{\mathbb{R}^n} u^4\, dx
 -\frac 16\int_{\mathbb{R}^n} (v_1^3+v_2^3)\,dx
 -\frac 12\int_{\mathbb{R}^n} u^2(\beta_1v_1+\beta_2v_2)\,dx \\
 \label{eq:N3}
\mathcal{N}=\{\mathbf{u}\in \mathbb{E}\setminus\{\mathbf{0}\}:  (\Phi'(\mathbf{u})| \mathbf{u})=0\},
\end{gather}

Let $U^*, V_j^*$ be the unique positive radially symmetric solutions of
$(-\Delta)^s u+\lambda_0 u=u^3$, $(-\Delta)^s v+\lambda_j v=\frac 12 v^2$
in $E$ respectively, $j=1,2$; see \cite{fl,fls}.

\begin{remark} \label{rem:semi-trivial3} \rm
The unique non-negative semi-trivial solutions of \eqref{NLS-KdV2-3} are given by
$\mathbf{v}_1^*=(0,V_1^*,0)$, $\mathbf{v}_2^*=(0,0,V^*_2)$ and $\mathbf{v}_{12}^*=(0,V_1^*,V^*_2)$.
\end{remark}

As in Section \ref{sec:pf2}, the first result about existence of ground states
 is the following theorem.

\begin{theorem}\label{th:3}
Assume $\beta_{j}>\Lambda_j$ for $j=1, 2$, then \eqref{NLS-KdV2-3} has a
positive radially symmetric ground state $\widetilde{\mathbf{u}}$.
\end{theorem}

\begin{proof}
We define
\begin{equation} \label{eq:Lambda-j}
\Lambda_j=\inf_{\varphi\in E\setminus\{0\}}
\frac{\|\varphi\|_0^2}{\int_{\mathbb{R}^n} V_{j}^*\varphi^2dx}\quad j=1,2.
\end{equation}
where $\|\cdot\|_0$ is the norm in $E$ with $\lambda_0$.

As in Proposition \ref{prop:5}-(ii), using that $\beta_{j}>\Lambda_j$,
$j=1,2$, one can show that both $\mathbf{v}_1^*$, $\mathbf{v}_2^*$ are saddle
points of the energy functional $\Phi$  (defined by \eqref{eq:Phi3})
constrained on the Nehari manifold $\mathcal{N}$ (defined by
\eqref{eq:Phi3}). Then
\begin{equation} \label{eq:energy3}
c_\mathcal{N}=\inf_{\mathcal{N}}\Phi<\min\{ \Phi(\mathbf{v}_1^*),\Phi (\mathbf{v}_2^*)\}<\Phi(\mathbf{v}_{12}^*)
=\Phi(\mathbf{v}_1^*)+\Phi (\mathbf{v}_2^*).
\end{equation}
By the Ekeland's variational principle, there exists a PS sequence
$\{\mathbf{u}_k\}_{k\in \mathbb{N}}\subset\mathcal{N}$, i.e.,
\begin{equation} \label{eq:PS1-3}
\Phi (\mathbf{u}_k)\to c_\mathcal{N}, \quad 
 \nabla_{\mathcal{N}}\Phi(\mathbf{u}_k)\to 0.
\end{equation}
The lack of compactness can be circumvent arguing in a similar way as in
the proof of Theorem \ref{th:1}, proving that for a subsequence,
$\mathbf{u}_k\rightharpoonup \widetilde{\mathbf{u}}$ weakly in $\mathbb{E}$ with 
$\widetilde{\mathbf{u}} \gvertneqq \mathbf{0}$,
 $\widetilde{\mathbf{u}}\in\mathcal{N}$ a critical point of $\Phi$ satisfying
$ \Phi(\widetilde{\mathbf{u}})=c_\mathcal{N}$, then $\widetilde{\mathbf{u}}$ is a non-negative ground state.

To prove the positivity of $\widetilde{\mathbf{u}}$, if one supposes that the
first component $u^* \equiv 0$, since the only non-negative
solutions of \eqref{NLS-KdV2-3} are the semi-trivial solutions
defined  in Remark \ref{rem:semi-trivial3}, we obtain a contradiction
with \eqref{eq:energy3}. Furthermore, if the second or third
component vanish then $\widetilde{\mathbf{u}}$ must be $\mathbf{0}$, and this is not
possible because $\Phi|_{\mathcal{N}}$ is bounded bellow by a positive
constant like in \eqref{eq:restriction}, then  $\mathbf{0}$ is an isolated
point of the set $\{\mathbf{u}\in \mathbb{E}\, :\: \Psi(\mathbf{u})=(\Phi'(\mathbf{u})|\mathbf{u})=0\}$, proving
that $\mathcal{N}$ is a complete manifold, as in the previous sections.
Then, the maximum principle shows that $\widetilde{\mathbf{u}}>\mathbf{0}.$ Finally,
to show that we have a radially symmetric ground state, we argue as in
the proof of Theorem \ref{th:1}.
\end{proof}

Furthermore, following the ideas in the proof of Theorem \ref{th:ground2} we
 have the following result.

\begin{theorem}\label{th:4}
 Assume that $\beta_1,\beta_2>0$ (but not necessarily $\beta_j>\Lambda_j$ as in
Theorem \ref{th:3}).
Then there exists a positive radially symmetric ground state $\widetilde{\mathbf{u}}$ provided
$\lambda_1,\lambda_2$ are sufficiently large.
\end{theorem}

\begin{proof}
The proof follows the same ideas as the one of Theorem
\ref{th:ground2} with appropriate changes. For example,  to prove the positivity,
one has to show that there exists $\mathbf{u}_0\in\mathcal{N}$
with $\Phi(\mathbf{u}_0)<\min\{ \Phi(\mathbf{v}_1^*),\Phi
(\mathbf{v}_2^*)\}$, that holds true provided $\lambda_1,\lambda_2$ are large enough.
We  omit details here.
\end{proof}

Plainly we can extend these results to systems with  an arbitrary number of
 equations $N>3$ as  follows,
\begin{equation} \label{eq:system-N}
\begin{gathered}
(-\Delta)^s u +\lambda_0 u  =   u^3+\sum_{k=1}^{N-1}\beta_{k}\, uv_k \\
(-\Delta)^s v_j +\lambda_j v_j  =  \frac 12 v_j^2+\frac 12\beta_{j} u^2; \quad
j=1,\cdots,N-1.
\end{gathered}
\end{equation}
Arguing as in  Theorems \ref{th:3} and \ref{th:4} we can show the next result.

\begin{theorem}\label{th:5}
There exists a positive radially symmetric ground state of \eqref{eq:system-N} if
\begin{itemize}
\item either
$$
\beta_{k}>\Lambda_k=\inf_{\varphi\in E\setminus\{
0\}}\frac{\|\varphi\|_0^2}{\int_{\mathbb{R}^n} V_{k}^*\varphi^2dx};\quad
k=1,\cdots N-1,
$$
where $V_k^*$ denotes the unique positive radial solution of
$\Delta v+\lambda_k v=\frac 12 v^2$ in $E$; $k=1,\cdots, N-1$,
\item or $\beta_j>0$ are arbitrary and  $\lambda_j$ are large enough;
 $j=1,\ldots,N-1$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk5} \rm
As was commented in \cite{c3} for the local setting, here in the nonlocal
fractional framework, another natural extension of \eqref{NLS-KdV2} to more than two
equations  different from \eqref{NLS-KdV2-3} is the following system
coming from 2NLFS-FKdV equations if $n=1$ or 3NLFS equations if $n=1,2,3$,
\begin{equation} \label{eq:NLS2-KdV}
\begin{gathered}
(-\Delta)^s u_1 +\lambda_1 u_1  =  u_1^3+\beta_{12} u_1u_2^2+\beta_{13}u_1v \\
(-\Delta)^s u_2 +\lambda_2 u_2  =   u_2^3+\frac 12\beta_{12} u_1^2u_2+\beta_{23} u_2v\\
(-\Delta)^s v+\lambda v  =  \frac 12v^2+\frac 12\beta_{13} u_1^2+\frac 12\beta_{23}u_2^2.
\end{gathered}.
\end{equation}
We denote $U_j$  the unique positive radially symmetric solution of
$(-\Delta)^s u +\lambda_j u  = u^3$ in $E$; $j=1, 2$;
and $V$ the corresponding positive radially symmetric solution to
 $(-\Delta)^s v +\lambda v = \frac 12 v^2$ in $E$.

Note that the non-negative radially symmetric semi-trivial solution $(0,0,V)$ is
a strict local minimum of the associated energy functional
constrained on the corresponding Nehari manifold provided
$$
\beta_{j3}<\Lambda_{j}=\inf_{\varphi\in E\setminus\{0\}}
\frac{\|\varphi\|_{\lambda_j}^2}{\int_{\mathbb{R}^n} V\varphi^2dx}\quad j=1,2.
$$
While if either $\beta_{13}>\Lambda_1$ or $\beta_{23}>\Lambda_2$ then
 $(0,0,V)$ is a saddle  point of $\Phi $ on $\mathcal{N}$.

There also exist semi-trivial solutions coming from the solutions
studied in Section \ref{sec:pf2}, with the first component or the
 second one $\equiv 0$. This fact makes different the analysis of
 \eqref{eq:NLS2-KdV} with respect to the previous studied systems
 \eqref{NLS-KdV2-3} and \eqref{eq:system-N}.
To finish, one could study more general extended systems of
\eqref{NLS-KdV2-3}, \eqref{eq:NLS2-KdV}
 with $N=m+\ell$; coming from $m$-NLFS and $\ell$-FKdV coupled equations with
$m,\ell\ge 2$ in the one dimensional case, or
 $N$-NLFS equations if $n=1,2,3$. Indeed, the existence of positive ground
states it is still unknown in the local setting ($s=1$)
 for this last kind of systems, including  \eqref{eq:NLS2-KdV} with $s=1$.
\end{remark}


\subsection*{Acknowledgments}
 The author is partially supported by the Ministry of Economy and Competitiveness
of Spain and FEDER under
Research Project MTM2016-80618-P, and by the INdAM - GNAMPA Project 2017
``Teoria e modelli non locali".

\begin{thebibliography}{00}

\bibitem{akanbook} N. Akhmediev, A. Ankiewicz;
\emph{Solitons, Nonlinear pulses and beams}. Champman \& Hall, London, 1997.

\bibitem{aa} J. Albert, J. Angulo Pava;
\emph{Existence and stability of ground-state solutions of a Schr\"odinger-KdV system}.
Proc. Roy. Soc. Edinburgh Sect. A, \textbf{133} (2003) 987-1029.

\bibitem{ab} J. Albert, S. Bhattarai;
\emph{Existence and stability of a two-parameter family of solitary waves
for an NLS-KdV system}.
Adv. Differential Equations,  \textbf{18}  (2013),  no. 11-12, 1129-1164.

\bibitem{ac1} A. Ambrosetti, E. Colorado;
\emph{Bound and ground states of coupled nonlinear Schr\"odinger equations.}
C. R. Math. Acad. Sci. Paris, \textbf{342} (2006), no. 7, 453-458.

\bibitem{ac2}  A. Ambrosetti, E. Colorado;
\emph{Standing waves of some coupled nonlinear Schr\"odin\-ger equations.}
J. Lond. Math. Soc., (\textbf{2}) 75 (2007), no. 1, 67-82.

\bibitem{acr} A. Ambrosetti, E. Colorado, D. Ruiz;
\emph{Multi-bump solitons to linearly coupled systems of nonlinear Schr\"odinger
equations.} Calc. Var. Partial Differential Equations, \textbf{30}
(2007), no. 1, 85-112.

\bibitem{Applebaum} D. Applebaum;
\emph{L{\'e}vy processes and stochastic calculus}. Second edition.
Cambridge Studies in Advanced
Mathematics, 116. Cambridge University Press, Cambridge, 2009.

\bibitem{ban} C. Bandle
;\emph{Isoperimetric inequalities and applications},
Monographs and Studies in Mathematics, \textbf{7}, Boston Pitman
Advanced Publishing Program 1980.

\bibitem{bt} T. Bartsch, Z.-Q. Wang;
\emph{Note on ground states of nonlinear Schr\"odinger systems.}
J. Partial Differential Equations, \textbf{19} (2006), no. 3, 200-207.

\bibitem{BL} H. Berestycki, P. L. Lions;
\emph{Nonlinear scalar field equations. I. Existence of a ground state}.
Arch. Rational Mech. Anal., \textbf{82} (1983), no. 4, 313–345.

\bibitem{Bertoin} J. Bertoin;
\emph{L{\'e}vy processes}. Cambridge Tracts in Mathematics,
121. Cambridge University Press, Cambridge, 1996.

\bibitem{brandle-colorado-depablo-sanchez}
C. Br\"{a}ndle, E. Colorado, A. de Pablo, U. S\'anchez;
\emph{A concave-convex elliptic problem involving the fractional Laplacian}.
Proc. Roy. Soc. Edinburgh Sect. A, \textbf{143} (2013), no. 1, 39-71.

\bibitem{cabre-sire} X. Cabr\'e, Y. Sire;
\emph{Nonlinear equations for fractional Laplacians, I:
Regularity, maximum principles, and Hamiltonian estimates.} Ann.
Inst. H. Poincar\'e Anal. Non Lin\'eaire,  \textbf{31}  (2014),  no.
1, 23-53.

\bibitem{Caffarelli-Silvestre} L. A. Caffarelli, L. Silvestre;
\emph{An extension problem related to the fractional Laplacian.}
Comm. Partial Differential Equations,  {\textbf32}  (2007), no. 7-9,
1245-1260.

\bibitem{chen-zou} Z. Chen; W. Zou;
\emph{An optimal constant for the existence of least energy solutions of a
coupled Schr\"odinger system.} Calc. Var. Partial Differential Equations,
\textbf{48} (2013), no. 3-4, 695–711.

\bibitem{c-fract}  E. Colorado;
\emph{Existence results for some systems of coupled fractional nonlinear
Schr\"odinger equations.}
 Recent trends in nonlinear partial differential equations. II.
Stationary problems, 135-150, Contemp. Math., \textbf{595},
  Amer. Math. Soc., Providence, RI, 2013.

\bibitem{c1}  E. Colorado;
\emph{Positive solutions to some systems of coupled nonlinear Schr\"odinger equations}.
Nonlinear Anal., \textbf{110} (2014) 104-112.

\bibitem{c2} E. Colorado;
\emph{Existence of bound and ground states for a system of coupled nonlinear
Schr\"odinger-KdV equations.}
C. R. Math. Acad. Sci. Paris, \textbf{353}  (2015),  no. 6, 511-516.

\bibitem{c3} E. Colorado;
\emph{On the existence of bound and ground states for some coupled nonlinear
Schr\"odinger--Korteweg-de Vries equations.}
Adv. Nonlinear Anal.,  \textbf{6}  (2017), no. 4, 407-426.

\bibitem{Cont-Tankov} R. Cont, P. Tankov;
\emph{Financial modelling with jump processes}.
Chapman \& Hall/CRC Financial Mathematics Series. Boca Raton, Fl,
2004.

\bibitem{cl} A. J. Corcho, F. Linares;
\emph{Well-posedness for the Schr\"odinger-Korteweg-de Vries system}.
Trans. Amer. Math. Soc., \textbf{359} (2007) 4089-4106.

\bibitem{co-oli-tav} S. Correia, F. Oliveira, H. Tavares;
\emph{semi-trivial vs. fully non-trivial ground states in cooperative cubic
Schr\"odinger systems with $d=3$  equations.} J. Funct. Anal.,  \textbf{271}
(2016), no. 8, 2247-2273.

\bibitem{dfo1} J.-P. Dias, M. Figueira, F. Oliveira;
 \emph{Existence of bound states for the coupled Schr\"odinger-KdV system with cubic
nonlinearity.} C. R. Math. Acad. Sci. Paris, \textbf{348} (2010),
no. 19-20, 1079-1082.

\bibitem{dfo2} J.-P. Dias, M. Figueira, F. Oliveira;
\emph{Well-posedness and existence of bound states for a coupled
Schr\"odinger-gKdV system.} Nonlinear Anal., \textbf{73} (2010), no.
8, 2686-2698.

\bibitem{dpv} S. Dipierro, G. Palatucci, E. Valdinoci;
 \emph{Existence and symmetry results for a Schr\"odinger type problem
involving the Fractional Laplacian.},  Matematiche (Catania)
\textbf{68},  (2013),  no. 1, 201-216.

\bibitem{eke}  I. Ekeland;
 \emph{On the variational principle}. J. Math. Anal. Appl.,
 \textbf{47} (1974), 324-353.

\bibitem{fl}  R. L. Frank,  E. Lenzmann;
 \emph{Uniqueness and Nonodegeneracy of Ground States for
$(-\Delta)^sQ + Q - Q^{a+1} = 0$ in $\mathbb{R}$}. To appear in Acta
Math.

\bibitem{fls} R. L. Frank,  E. Lenzmann, L. Silvestre;
 \emph{Uniqueness of radial solutions for the fractional Laplacian.}
Comm. Pure Appl. Math., \textbf{69}  (2016),  no. 9, 1671-1726.

\bibitem{fo} M. Funakoshi, M. Oikawa;
\emph{The resonant Interaction between a Long Internal Gravity Wave and a
Surface Gravity Wave Packet}. J. Phys. Soc. Japan,
 \textbf{52} (1983), no.1, 1982-1995.

\bibitem{fmm}  N. Fusco, V. Millot, M. Morini;
 \emph{A quantitative isoperimetric inequality for fractional perimeters.}
J. Funct. Anal., \textbf{261} (2011), no. 3, 697-715.

\bibitem{Landkof} N. S. Landkof;
\emph{Foundations of modern potential theory}.
Die Grundlehren der mathematischen Wissenschaften,
Band 180. Springer-Verlag, New York-Heidelberg, 1972.

\bibitem{Laskin} N. Laskin;
 \emph{Fractional Schr\"odinger equation}. Phys. Rev. E,
 (3)  \textbf{66}  (2002),  no. 5, 056108, 7 pp.

\bibitem{linwei} T-C. Lin, J. Wei;
\emph{Ground state of $N$ coupled nonlinear Schr\"odinger  equations in $\mathbb{R}^n$,
$n\leq 3$}. Comm. Math. Phys., \textbf{255} (2005), 629-653.

\bibitem{Lions-JFA82} P. L. Lions;
\emph{Sym\'etrie et compacit\'e dans les espaces de Sobolev}.
J. Funct. Anal., \textbf{49} (1982), no. 3, 315-334.

\bibitem{liu-wang} Z. Liu, Z.-Q. Wang;
\emph{Ground states and bound states of a nonlinear Schr\"odinger system.}
 Adv. Nonlinear Stud. \textbf{10} (2010), no. 1, 175-193.

\bibitem{mmp} L. Maia, E. Montefusco, B. Pellacci;
\emph{Positive solutions for a weakly coupled nonlinear Schr\"odinger system.}
J. Differential Equations \textbf{229} (2006), no. 2, 743-767.

\bibitem{pomp} A. Pomponio;
 \emph{Coupled nonlinear Schr\"odinger systems with potentials}.
J. Differential Equations, \textbf{227} (2006), no. 1, 258-281.

\bibitem{sirakov} B. Sirakov;
\emph{Least energy solitary waves for a system of nonlinear Schr\"odinger
equations in $\Bbb R^n$.} Comm. Math. Phys., \textbf{271} (2007), no. 1, 199-221.

\bibitem{soave-tavares} N. Soave, H. Tavares;
\emph{New existence and symmetry results for least energy positive solutions
of Schr\"odinger systems with mixed competition and cooperation terms.} J.
Differential Equations,  \textbf{261}  (2016),  no. 1, 505-537.

\bibitem{Stein} E. M. Stein;
\emph{Singular integrals and differentiability properties of functions}.
 Princeton Mathematical Series, No. 30 Princeton University Press, Princeton,
 N.J. 1970.

\bibitem{stroock} D. W. Stroock;
\emph{An introduction to the theory of large deviations}.
 Universitext. Springer, New York, 1984.

\bibitem{varop} N. Th. Varopoulos;
\emph{Hardy-Littlewood theory for semigroups.} J.
Funct. Anal., \textbf{63} (1985), no. 2, 240–260.

\bibitem{Willem} M. Willem;
\emph{Minimax Theorems}, Progr. Nonlinear Differential Equations Appl., vol. 24,
 Birkh\"auser, Boston, 1996.

\end{thebibliography}


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