# Normalized ground states for a coupled Schrödinger system: Mass super-
critical case

Louis Jeanjean  ,  Jianjun Zhang  and  Xuexiu Zhong
Université de Franche-Comté
CNRS, UMR 6623, LmB, F-25000 Besançon, France<EMAIL_ADDRESS>
College of Mathematics and Statistics
Chongqing Jiaotong University
Chongqing 400074, PR China<EMAIL_ADDRESS>
South China Research Center for Applied Mathematics and Interdisciplinary
Studies
South China Normal University
Guangzhou 510631, PR China<EMAIL_ADDRESS>
###### Abstract.

We consider the existence of solutions
$(\lambda_{1},\lambda_{2},u,v)\in\mathbb{R}^{2}\times(H^{1}(\mathbb{R}^{N}))^{2}$
to systems of coupled Schrödinger equations

$\begin{cases}-\Delta u+\lambda_{1}u=\mu_{1}u^{p-1}+\beta
r_{1}u^{r_{1}-1}v^{r_{2}}\quad&\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\\\ -\Delta v+\lambda_{2}v=\mu_{2}v^{q-1}+\beta
r_{2}u^{r_{1}}v^{r_{2}-1}\quad&\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\\\ 0<u,v\in H^{1}(\mathbb{R}^{N}),\,1\leq N\leq
4,&\end{cases}$

satisfying the normalization

$\int_{\mathbb{R}^{N}}u^{2}\mathrm{d}x=a\quad\mbox{and}\quad\int_{\mathbb{R}^{N}}v^{2}\mathrm{d}x=b.$

Here $\mu_{1},\mu_{2},\beta>0$ and the prescribed masses $a,b>0$. We focus on
the coupled purely mass super-critical case, i.e.,

$2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$

with $2^{*}$ being the Sobolev critical exponent, defined by $2^{*}:=+\infty$
for $N=1,2$ and $2^{*}:=\frac{2N}{N-2}$ for $N=3,4$. We optimize the range of
$(a,b,\beta,r_{1},r_{2})$ for the existence. In particular, for $N=3,4$ with
$r_{1},r_{2}\in(1,2)$, our result indicates the existence for all $a,b>0$ and
$\beta>0$.

###### Key words and phrases:

Schrödinger equation; positive normalized solution; global branch.

Xuexiu Zhong NSFC was supported by the NSFC (No.12271184), Guangdong Basic and
Applied Basic Research Foundation (2021A1515010034),Guangzhou Basic and
Applied Basic Research Foundation(202102020225). Jianjun Zhang was supported
by the NSFC (No.12371109)

##  1\. Introduction

Time-dependent systems of coupled nonlinear Schrödinger equations of the form

$\begin{cases}-i\frac{\partial}{\partial
t}\Phi_{1}=\Delta\Phi_{1}+g_{1}(|\Phi_{1}|^{2})\Phi_{1}+\partial_{1}\varphi(|\Phi_{1}|^{2},|\Phi_{2}|^{2})\Phi_{1},\\\
-i\frac{\partial}{\partial
t}\Phi_{2}=\Delta\Phi_{2}+g_{2}(|\Phi_{2}|^{2})\Phi_{2}+\partial_{2}\varphi(|\Phi_{1}|^{2},|\Phi_{2}|^{2})\Phi_{2},\\\
\Phi_{j}=\Phi_{j}(x,t)\in\mathbb{C},\,j=1,2,N\geq
1,\end{cases}\quad(x,t)\in\mathbb{R}^{N}\times\mathbb{R},$ (1.1)

are used as model for various physical phenomena, for instance binary mixtures
of Bose-Einstein condensates, or the propagation of mutually incoherent wave
packets in nonlinear optics; see e.g. [1, 2, 9, 18]. The ansatz
$\Phi_{j}(x,t)=e^{i\lambda_{j}t}u_{j}(x),j=1,2$ for solitary wave solutions
leads to the elliptic system

$\begin{cases}-\Delta
u_{1}+\lambda_{1}u_{1}=f_{1}(u_{1})+\partial_{1}H(u_{1},u_{2}),\\\ -\Delta
u_{2}+\lambda_{2}u_{2}=f_{2}(u_{2})+\partial_{2}H(u_{1},u_{2}),\end{cases}\quad\hbox{in}\;\mathbb{R}^{N},$
(1.2)

with $f_{j}(u_{j})=g_{j}(|u_{j}|^{2})u_{j},j=1,2$ and
$H(u_{1},u_{2})=\frac{1}{2}\varphi(|u_{1}|^{2},|u_{2}|^{2})$.

Since the masses

$\int_{\mathbb{R}^{N}}|\Phi_{j}|^{2}\mathrm{d}x,j=1,2$

are preserved along trajectories of (1.1), it is natural to consider them as
prescribed. A natural approach to obtain solutions to (1.2) satisfying the
normalization constraints

$\int_{\mathbb{R}^{N}}|u_{j}|^{2}=a_{j},j=1,2$ (1.3)

consists in finding critical points $(u_{1},u_{2})\in
H^{1}(\mathbb{R}^{N},\mathbb{R}^{2})$ of the energy functional

$\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|\nabla u_{1}|^{2}+|\nabla
u_{2}|^{2}\right)\mathrm{d}x-\int_{\mathbb{R}^{N}}\left(F_{1}(u_{1})+F_{2}(u_{2})+H(u_{1},u_{2})\right)\mathrm{d}x$

under the constraint (1.3), where $F_{j}(s):=\int_{0}^{s}f_{j}(\tau)d\tau$ is
the primitive function of $f_{j},j=1,2$. Then the parameters
$\lambda_{1},\lambda_{2}$ appear as Lagrange multipliers.

In the present paper we focus on the existence of solutions
$(\lambda_{1},\lambda_{2},u,v)\in\mathbb{R}^{2}\times
H^{1}(\mathbb{R}^{N},\mathbb{R}^{2})$ to the specific problem

$\begin{cases}-\Delta u+\lambda_{1}u=\mu_{1}|u|^{p-2}u+\beta
r_{1}|u|^{r_{1}-2}u|v|^{r_{2}}\quad&\text{in $\mathbb{R}^{N}$,}\\\ -\Delta
v+\lambda_{2}v=\mu_{2}|v|^{q-2}v+\beta
r_{2}|u|^{r_{1}}|v|^{r_{2}-2}v\quad&\text{in $\mathbb{R}^{N}$,}\\\
\int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d}x=a,\,\int_{\mathbb{R}^{N}}|v|^{2}\mathrm{d}x=b.\end{cases}$
(1.4)

Here $\mu_{1},\mu_{2},\beta>0$ are prescribed as are the masses $a,b>0$. The
corresponding energy functional $J:\mathcal{H}\mapsto\mathbb{R}$ is defined by

$J[u,v]=\frac{1}{2}[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]-\frac{\mu_{1}}{p}\|u\|_{p}^{p}-\frac{\mu_{2}}{q}\|v\|_{q}^{q}-\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x,$

where $\mathcal{H}:=H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})$ is
equipped with the norm

$\|[u,v]\|_{\mathcal{H}}:=\left(\|u\|_{H^{1}}^{2}+\|v\|_{H^{1}}^{2}\right)^{\frac{1}{2}}.$

The normalization constraint being $S_{a}\times S_{b}$ where

$S_{a}:=\left\\{u\in
H^{1}(\mathbb{R}^{N}):\|u\|_{2}^{2}=a\right\\}\quad\mbox{and}\quad
S_{b}:=\left\\{u\in H^{1}(\mathbb{R}^{N}):\|u\|_{2}^{2}=b\right\\}.$

It is standard to show that $J:\mathcal{H}\mapsto\mathbb{R}$ is of class
$C^{1}$ and that $J^{\prime}$ takes bounded sets to bounded sets.

For the case $N=3,r_{1}=r_{2}=2,\mu_{1}=\mu_{2},p=q=4$, assuming that
$\beta<0$, the existence of infinitely many positive solutions (namely $u>0$
and $v>0$) to Problem (1.4) was obtained by Bartsch and Soave in [5]. When
$\beta>0$, Gou and Jeanjean in [11] obtain a multiplicity result. More
precisely, under one of the following conditions

  * $(H_{0})$

$N\geq 1,1<p,q<2+\frac{4}{N},r_{1},r_{2}>1,2+\frac{4}{N}<r_{1}+r_{2}<2^{*}$;

  * $(H_{1})$

$N\geq 1,2+\frac{4}{N}<p,q<2^{*},r_{1},r_{2}>1,r_{1}+r_{2}<2+\frac{4}{N}$,

two couples of solutions to Problem (1.4) are obtained. One is a local
minimizer for $J$ constrained on $S_{a}\times S_{b}$, the other is obtained
through a constrained mountain pass or a constrained linking. In both cases it
is assumed that $0<\beta<\beta_{0}$ is small enough.

For the pure mass super-critical case, i.e.,
$2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$, Bartsch, Jeanjean and Soave in [4]
consider the case of $N=3,p=q=4,r_{1}=r_{2}=2$. They obtain the existence of
positive solutions to Problem (1.4) provided $0<\beta<\beta_{1}(a,b)$ or
$\beta>\beta_{2}(a,b)$. A more general case is considered by Bartsch and
Jeanjean in [3] but it is also required there that $0<\beta<\beta_{1}(a,b)$ or
$\beta>\beta_{2}(a,b)$. The values $\beta_{1}(a,b)$ and $\beta_{2}(a,b)$
depend on $a,b$ heavily. Typically, $\beta_{1}(a,b)\rightarrow 0$ as
$a/b\rightarrow 0$ or $a/b\rightarrow+\infty$, while
$\beta_{2}(a,b)\rightarrow+\infty$ as $a/b\rightarrow 0$ or
$a/b\rightarrow+\infty$. In particular, there is no value of $\beta>0$ so that
the results of [4, 3] yield a solution to Problem (1.4) for all masses.

In view of the above considerations, it is natural to investigate whether
there exists some $\beta>0$ for which the existence of solutions is valid for
all masses. Actually, this issue was already raised as an open problem by
Bartsch, Jeanjean and Soave, see [4, Remark 1.3-(a)and (d)]. In [6], Bartsch,
Zhong and Zou gave a first answer. Precisely, assuming that
$N=3,p=q=4,r_{1}=r_{2}=2$, they develop a new approach based on fixed point
index in cones, bifurcation theory, and a continuation method to deal with
Problem (1.4). For the range of
$\beta\in(0,\tau_{0}\min\\{\mu_{1},\mu_{2}\\}]\cup(\tau_{0}\max\\{\mu_{1},\mu_{2}\\},+\infty)$,
where $\tau_{0}\in(0,1)$ is explicitly known, the existence result holds for
all masses. For the range of
$\beta\in(\tau_{0}\min\\{\mu_{1},\mu_{2}\\},\tau_{0}\max\\{\mu_{1},\mu_{2}\\}]$
whether the existence result holds depends on the masses $a$ and $b$, more
precisely, on the ratio $a/b$. For example, when $\mu_{2}<\mu_{1}$, for $a/b$
small, the existence result is satisfied. On the contrary when
$\mu_{2}\leq\beta\leq\tau_{0}\mu_{1}$, there is no positive normalized
solution provided $a/b$ is large.

Our aim in the present paper is to make further advances on Bartsch-Jeanjean-
Soave’s open problem. Namely to explicit new ranges of $\beta>0$ for which
there exist a solution to Problem (1.4) for any $a>0$ and $b>0$. To appreciate
our results observe that in [6] all the nonlinear terms share the same
homogeneity. In our results $p,q,r_{1}+r_{2}$ are not necessary the same.

######  Remark 1.1.

For the fixed frequencies problem (that is when $\lambda_{1},\lambda_{2}$ are
given) when $N=3,r_{1}=r_{2}=2,\mu_{1}=\mu_{2},p=q=4$, usually one also need
some structural conditions on the parameters to guarantee the existence of
positive solutions. In that case there is also an open problem, which is
called Sirakov’s open problem in the literature, concerning the determination
of the best range for the existence of positive solutions. In that direction
we refer to [16, Remark 4] and [19].

##  2\. Hypotheses and statement of results

In order to state our main results we first recall some classical facts.

For $p\in(2,2^{*})$, it is well known that the following scalar problem

$\begin{cases}-\Delta u+u=u^{p-1}\quad\;&\hbox{in}\;\mathbb{R}^{N},\\\
u>0\quad&\hbox{in}\leavevmode\nobreak\ \mathbb{R}^{N},\\\
u(0)=\max_{x\in\mathbb{R}^{N}}u(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\hbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ u\in
H^{1}(\mathbb{R}^{N})\end{cases}$ (2.1)

has a unique solution, denoted by $U_{p}$, which is a radial function, cf.
[13].

For $a,\mu\in\mathbb{R}^{+}$ fixed, consider the solutions
$(\lambda,\omega)\in\mathbb{R}\times H^{1}(\mathbb{R}^{N})$ with $\lambda>0$,
of

$\begin{cases}-\Delta\omega+\lambda\omega=\mu\,\omega^{p-1}\;&\hbox{in}\;\mathbb{R}^{N},\\\
w>0\quad&\hbox{in}\leavevmode\nobreak\ \mathbb{R}^{N},\\\
\omega(0)=\max_{x\in\mathbb{R}^{N}}\omega(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \hbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ u\in
H^{1}(\mathbb{R}^{N}),\\\ \|\omega\|_{2}^{2}=a.\end{cases}$ (2.2)

It is also classical that these solutions are unique and can be obtained by
scaling the solution to (2.1). Precisely, for $p\neq 2+\frac{4}{N}$, the
couple

$\lambda=\lambda_{p,\mu,a}:=\mu^{-\frac{4}{N(p-2)-4}}\|U_{p}\|_{2}^{\frac{4(p-2)}{N(p-2)-4}}a^{-\frac{2(p-2)}{N(p-2)-4}}>0$
(2.3)

and

$U_{p,\mu,\lambda}(x):=\left(\frac{\lambda}{\mu}\right)^{\frac{1}{p-2}}U_{p}(\sqrt{\lambda}x),$

solves (2.2). We denote this unique solution by

$z_{p,\mu,a}(x):=U_{p,\mu,\lambda_{p,\mu,a}}(x).$ (2.4)

######  Definition 2.1.

Let $a>0$, $b>0$ be given. We call $[u,v]$ a ground state solution (or energy
ground state) to Problem (1.4) if $[u,v]$ is a solution to Problem (1.4) and

$J[u,v]=\inf\left\\{J[\phi,\psi]:[\phi,\psi]\leavevmode\nobreak\ \hbox{is a
solution to Problem \eqref{eq:20220902-maine1}}\right\\}.$

Finally, let $p,q,\mu_{1},\mu_{2}$ be fixed and for $a>0$ define

$\displaystyle b_{p,q,\mu_{1},\mu_{2},a}:=$
$\displaystyle\left(\frac{\mu_{2}}{\mu_{1}}\frac{(q-2)N-4}{(p-2)N-4}\frac{2N-(N-2)p}{2N-(N-2)q}\right)^{\frac{(q-2)N-4}{2N-(N-2)q}}\|U_{q}\|_{2}^{\frac{4(q-2)}{2N-(N-2)q}}$
$\displaystyle\times\|U_{p}\|_{2}^{-\frac{4(p-2)}{2N-(N-2)q}\frac{(q-2)N-4}{(p-2)N-4}}a^{\frac{2N-(N-2)p}{2N-(N-2)q}\frac{(q-2)N-4}{(p-2)N-4}}.$

and

$\beta_{p,\mu,a,N,r}:=\frac{1}{2}\inf_{h\in
H^{1}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\|\nabla
h\|_{2}^{2}}{\int_{\mathbb{R}^{N}}|z_{p,\mu,a}|^{r}|h|^{2}\mathrm{d}x}.$ (2.5)

We remark that $J[0,z_{q,\mu_{2},b}]<J[z_{p,\mu_{1},a},0]$ if and only if
$b>b_{p,q,\mu_{1},\mu_{2},a}$ (by a direct calculation from Lemma 3.2-(i)).
Now we can state our main result.

######  Theorem 2.2.

Let $1\leq N\leq 4,2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$ and $r_{1},r_{2}>1$.
Then the following holds true.

  * (i)

For $a>0$ and $b\in[b_{p,q,\mu_{1},\mu_{2},a},+\infty)$. If

$\begin{cases}r_{1}<2\\\ \beta>0\end{cases}\leavevmode\nobreak\
\hbox{or}\leavevmode\nobreak\ \begin{cases}r_{1}=2\\\
\beta>\beta_{q,\mu_{2},b,N,r_{2}}\,,\end{cases}$

then there exists a ground state solution $(\lambda_{1},\lambda_{2},u,v)$ to
Problem (1.4). In addition, $\lambda_{1}>0,\lambda_{2}>0$ and $u,v$ are
Schwartz symmetric functions.

  * (ii)

Let $a>0$ and $b\in(0,b_{p,q,\mu_{1},\mu_{2},a}]$. If

$\begin{cases}r_{2}<2\\\ \beta>0\end{cases}\leavevmode\nobreak\
\hbox{or}\leavevmode\nobreak\ \begin{cases}r_{2}=2\\\
\beta>\beta_{p,\mu_{1},a,N,r_{1}}\,,\end{cases}$

then there exists a ground state solution $(\lambda_{1},\lambda_{2},u,v)$ to
Problem (1.4). In addition, $\lambda_{1}>0,\lambda_{2}>0$ and $u,v$ are
Schwartz symmetric functions.

In Lemmas 7.1 and 7.2, the behavior of $\beta_{p,\mu,a,N,r}$ with respect to
the various parameters is studied. In particular, for the case of $N=1,2$, we
prove that $\beta_{p,\mu,a,N,r}=0$ always is true, so we have the following
conclusion.

######  Corollary 2.3.

For $N=1,2$, assume that $2+\frac{4}{N}<p,q$ and
$r_{1}>1,r_{2}>1,r_{1}+r_{2}>2+\frac{4}{N}$. Then

  * (i)

for any $a>0$ and $\beta>0$, there exists a ground state solution
$(\lambda_{1},\lambda_{2},u,v)$ to Problem (1.4) provided
$b\in[b_{p,q,\mu_{1},\mu_{2},a},+\infty)$ and $r_{1}\leq 2$. In addition,
$\lambda_{1}>0,\lambda_{2}>0$ and $u,v$ are Schwartz symmetric functions.

  * (ii)

for any $a>0$ and $\beta>0$, there exists a positive ground state solution
$(\lambda_{1},\lambda_{2},u,v)$ to Problem (1.4) provided
$b\in(0,b_{p,q,\mu_{1},\mu_{2},a}]$ and $r_{2}\leq 2$. In addition,
$\lambda_{1}>0,\lambda_{2}>0$ and $u,v$ are Schwartz symmetric functions.

For $N=3,4$, we get directly from Theorem 2.2.

######  Corollary 2.4.

For $N=3$ or $N=4$, assume that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$ and
$1<r_{1},r_{2}<2$. Then for any $a,b,\beta>0$, there exists a ground state
solution $(\lambda_{1},\lambda_{2},u,v)$ to Problem (1.4). In addition,
$\lambda_{1}>0,\lambda_{2}>0$ and $u,v$ are Schwartz symmetric functions.

Let us now describe the main steps of the proof of Theorem 2.2.

The ground states will be obtain as minimum of the functional $J$ on the
natural constraint

$\mathcal{P}:=\left\\{[u,v]\in\mathcal{H}\backslash\\{[0,0]\\}:P[u,v]=0\right\\},$
(2.6)

where

$\displaystyle P[u,v]:=$ $\displaystyle\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}-\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}-\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}$
$\displaystyle-\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x.$

Here also one can show, in a standard way, that $P$ is of class $C^{1}$ and
that $P^{\prime}$ takes bounded sets into bounded sets. This constraint is a
natural one in the sense that, as a consequence of a Pohozaev identity, all
critical points of $J$ constrained to $S_{a}\times S_{b}$ belong to
$\mathcal{P}$. Thus, if we manage to show that

$\inf_{\mathcal{P}\cap(S_{a}\times S_{b})}J[u,v]$

is achieved by a critical point this will prove the existence of a ground
state.

Instead of attacking directly this problem we consider a relaxed one. For
$a\geq 0$, we introduce the set

$D_{a}:=\\{u\in H^{1}(\mathbb{R}^{N}):\|u\|_{2}^{2}\leq a\\}$

and for any $(a,b)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$, we define

$\mathcal{P}_{(a,b)}:=\mathcal{P}\cap(D_{a}\times D_{b}).$

The relaxed problem is :

to find a critical point
$\displaystyle[u,v]\in\mathcal{H}\backslash\\{[0,0]\\}\mbox{ of }J\mbox{
constrained to }D_{a}\times D_{b}$ $\displaystyle\mbox{ at the level
}C_{(a,b)}:=\inf_{\mathcal{P}_{(a,b)}}J[u,v].$ (2.7)

Once this problem is settle, we show, see Lemma 8.2, that $[u,v]\in
S_{a}\times S_{b}$ solving thus the initial problem.

This idea of first considering a relaxed problem is reminiscent of the works
[7, 15]. However the strategy of the proofs in [7, 15] and in the present work
are essentially distinct.

To solve Problem (2), we shall make use of a classical result due to
Ghoussoub. This result [10, Theorem 4.1] provides a sequence
$\\{(u_{n},v_{n})\\}\subset\mathcal{H}$ for $J$ restricted to $D_{a}\times
D_{b}$ at the level $C_{(a,b)}$ which have useful additional properties. In
particular it is localized around $\mathcal{P}$ and this insure its
boundedness.

To obtain the convergence of $\\{(u_{n},v_{n})\\}\subset\mathcal{H}$, and thus
the existence of a critical point at this level, a key point is to show that
the level $C_{(a,b)}$ is sufficiently low. More precisely, observe that
$[z_{p,\mu_{1},a},0],[0,z_{q,\mu_{2},b}]\in\mathcal{P}_{(a,b)}$, and thus that
it holds

$C_{(a,b)}\leq\min\\{m_{p,\mu_{1},a},m_{q,\mu_{2},b}\\},$

where we have set

$m_{p,\mu_{1},a}:=J(z_{p,\mu_{1},a},0)\quad\mbox{and}\quad
m_{q,\mu_{2},b}:=J(0,z_{q,\mu_{2},b}).$

The condition that we use to conclude to the convergence of
$\\{(u_{n},v_{n})\\}\subset\mathcal{H}$ is

$C_{(a,b)}<\min\\{m_{p,\mu_{1},a},m_{q,\mu_{2},b}\\}.$ (2.8)

To prove (2.8) we study the behavior of $J$ restricted to
$\mathcal{P}_{(a,b)}$ locally around $[z_{p,\mu_{1},a},0]$ and
$[0,z_{q,\mu_{2},b}]$ in a way which seems new to us. In particular the
inequality (2.8) is proved testing $J$ on a point in
$\mathcal{P}_{(a,b)}\backslash\\{S_{a}\times S_{b}\\}$. It is at this step,
and only there, that appears the need to restrict our results to small values
of $r_{1}>0$ and $r_{2}>0$. Having proved (2.8) the convergence of
$\\{(u_{n},v_{n})\\}\subset\mathcal{H}$ follows, leading to the existence of a
ground state.

To obtain the additional properties of the ground states stated in Theorem
2.2, and its corollaries, we use the fact that any minimum for

$\inf_{\mathcal{P}\cap(S_{a}\times S_{b})}J[u,v]$

is a critical point for $J$ constrained to $S_{a}\times S_{b}$ and that it is
achieved by Schwartz symmetric functions.

The paper is organized as follows. In Section 3 we study the scalar problems
which appear in the discussion of the convergence of our Palais-Smale sequence
$\\{(u_{n},v_{n})\\}\subset\mathcal{H}$. Section 4 is devoted to the analysis
of the geometry of the set $\mathcal{P}_{(a,b)}$. We also prove that any
critical point for $J$ restricted to $\mathcal{P}_{(a,b)}$ is a critical point
for $J$ restricted to $D_{a}\times D_{b}$, namely that no Lagrange multiplier
arises in the presence of the constraint $\mathcal{P}$. In Section 5 we show
that it is not restrictive to search ground states within the Schwartz
symmetric functions. In Section 6 we establish the existence of our particular
Palais-Smale sequence $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$. Section 7 is
fully devoted to the proof of the strict inequality (2.8). Finally, in Section
8, we establish the convergence of $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$ and
we give the proof of Theorem 2.2.

##  3\. Some Preliminaries

We present in this section results which are slightly more general than
needed.

Let us introduce the set

$\tilde{\mathcal{P}}_{p,\mu}:=\left\\{u\neq 0:\|\nabla
u\|_{2}^{2}=\frac{(p-2)N}{2p}\mu\|u\|_{p}^{p}\right\\}$

and for $a>0$,

$\tilde{\mathcal{P}}_{p,\mu,a}:=\tilde{\mathcal{P}}_{p,\mu}\cap
S_{a}=\left\\{\|u\|_{2}^{2}=a:\|\nabla
u\|_{2}^{2}=\frac{(p-2)N}{2p}\mu\|u\|_{p}^{p}\right\\}.$

Recalling the fiber map

$u(x)\mapsto(t\star u)(x):=t^{\frac{N}{2}}u(tx),$ (3.1)

for $(t,u)\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$, which preserves the
$L^{2}$-norm. We have the following results.

######  Lemma 3.1.

  * (i)

If $u$ is a solution to (2.2), then $u\in\tilde{\mathcal{P}}_{p,\mu,a}$. Also
the unique solution to (2.2) minimizes $I_{p,\mu}$ on
$\tilde{\mathcal{P}}_{p,\mu,a}$.

  * (ii)

If $p\neq 2+\frac{4}{N}$, then for any $u\neq 0$, there exists a unique
$t=t(u)>0$ such that $t\star u\in\tilde{\mathcal{P}}_{p,\mu}$ and it holds
that

$I_{p,\mu}[t\star u]=\begin{cases}\max_{s>0}I_{p,\mu}[s\star
u]\quad&\hbox{if}\leavevmode\nobreak\ 2+\frac{4}{N}<p<2^{*},\\\
\min_{s>0}I_{p,\mu}[s\star u]\quad&\hbox{if}\leavevmode\nobreak\
2<p<2+\frac{4}{N}.\end{cases}$ (3.2)

  * (iii)

Let $z_{p,\mu,a}$ be defined by (2.4). Then

$\displaystyle m_{p,\mu,a}:=$ $\displaystyle
I_{p,\mu}[z_{p,\mu,a}]=\inf_{u\in\tilde{\mathcal{P}}_{p,\mu,a}}I_{p,\mu}[u]$
$\displaystyle=$ $\displaystyle\begin{cases}\inf_{u\in
S_{a}}\max_{s>0}I_{p,\mu}[s\star u]\quad&\hbox{if}\leavevmode\nobreak\
2+\frac{4}{N}<p<2^{*},\\\ \inf_{u\in S_{a}}\min_{s>0}I_{p,\mu}[s\star
u]=\inf_{u\in S_{a}}I_{p,\mu}[u]\quad&\hbox{if}\leavevmode\nobreak\
2<p<2+\frac{4}{N}.\end{cases}$

###### Proof.

(i) We refer to [4, Lemma 2.1].
(ii) For $u\neq 0$ and $s>0$, a direct calculation shows that

$f(s):=I_{p,\mu}[s\star u]=\frac{1}{2}\|\nabla
u\|_{2}^{2}s^{2}-\frac{\mu}{p}\|u\|_{p}^{p}s^{\frac{(p-2)N}{2}}.$

It is easy to see that $s\star u\in\tilde{\mathcal{P}}_{p,\mu}$ if and only if
$s$ is a critical point of $f(s)$. Then if $p\neq 2+\frac{4}{N}$, one can show
that $f(s)$ has an unique critical point

$t=t(u):=\left(\frac{\|\nabla
u\|_{2}^{2}}{\frac{(p-2)N}{2p}\mu\|u\|_{p}^{p}}\right)^{\frac{2}{(p-2)N-4}}$

and

$f(t)=\frac{(p-2)N-4}{4p}\left(\frac{2p}{(p-2)N}\right)^{\frac{(p-2)N}{(p-2)N-4}}\left(\|\nabla
u\|_{2}^{2}\right)^{\frac{(p-2)N}{(p-2)N-4}}\left(\mu\|u\|_{p}^{p}\right)^{\frac{4}{(p-2)N-4}}.$

Furthermore, $f(t)>0$ achieves the maximum if $p>2+\frac{4}{N}$, while
$f(t)<0$ achieves the minimum if $p<2+\frac{4}{N}$. Hence, (3.2) holds.
(iii) By (i) and (ii), it is easy to see. ∎

######  Lemma 3.2.

  * (i)

If $2+\frac{4}{N}<p<2^{*}$, then $m_{p,\mu,a}$ is continuous and decreases
strictly respect to $a\in\mathbb{R}^{+}$, furthermore,

$\lim_{a\rightarrow
0^{+}}m_{p,\mu,a}=+\infty\;\hbox{and}\;\lim_{a\rightarrow+\infty}m_{p,\mu,a}=0.$

  * (ii)

If $2<p<2+\frac{4}{N}$, then $m_{p,\mu,a}$ is continuous and decreases
strictly with respect to $a\in\mathbb{R}^{+}$, furthermore,

$\lim_{a\rightarrow
0^{+}}m_{p,\mu,a}=0\;\hbox{and}\;\lim_{a\rightarrow+\infty}m_{p,\mu,a}=-\infty.$

###### Proof.

By Lemma 3.1-(iii), $m_{p,\mu,a}=I_{p,\mu}[z_{p,\mu,a}]$. A direct calculation
shows that

$m_{p,\mu,a}=\frac{1}{2}\frac{(p-2)N-4}{2N-(N-2)p}\left(\|U_{p}\|_{2}^{2}\right)^{\frac{2(p-2)}{(p-2)N-4}}\mu
a^{-\frac{2N-(N-2)p}{(p-2)N-4}}.$

If $2+\frac{4}{N}<p<2^{*}$, we see the exponent
$-\frac{2N-(N-2)p}{(p-2)N-4}<0$, hence $m_{p,\mu,a}$ decreases strictly with
respect to $a\in\mathbb{R}^{+}$. Furthermore,

$\lim_{a\rightarrow
0^{+}}m_{p,\mu,a}=+\infty\;\hbox{and}\;\lim_{a\rightarrow+\infty}m_{p,\mu,a}=0.$

Similarly, if $2<p<2+\frac{4}{N}$, then $-\frac{2N-(N-2)p}{(p-2)N-4}>0$, and
noting that $\frac{1}{2}\frac{(p-2)N-4}{2N-(N-2)p}<0$ in such a case, thus
$m_{p,\mu,a}$ also decreases strictly respect to $a\in\mathbb{R}^{+}$.
Furthermore,

$\lim_{a\rightarrow
0^{+}}m_{p,\mu,a}=0\;\hbox{and}\;\lim_{a\rightarrow+\infty}m_{p,\mu,a}=-\infty.$

∎

##  4\. A Pohozaev type constraint and its properties

In this section we establish various properties of the constraint
$\mathcal{P}$ and we prove, in particular, it does not generate a Lagrange
parameter. First observe that,

######  Lemma 4.1.

Let $\mathcal{P}$ be the so-called Pohozaev manifold defined by (2.6). If
$[u,v]\in\mathcal{H}\backslash\\{[0,0]\\}$ is a solution to

$\begin{cases}-\Delta u+\lambda_{1}u=\mu_{1}|u|^{p-2}u+\beta
r_{1}|u|^{r_{1}-2}u|v|^{r_{2}}\quad&\text{in $\mathbb{R}^{N}$,}\\\ -\Delta
v+\lambda_{2}v=\mu_{2}|v|^{q-2}v+\beta
r_{2}|u|^{r_{1}}|v|^{r_{2}-2}v\quad&\text{in $\mathbb{R}^{N}$,}\end{cases}$
(4.1)

then $[u,v]\in\mathcal{P}$.

###### Proof.

This result can be shown as in [4, Lemma 4.6]. ∎

Now, for any fixed $[u,v]\in\mathcal{H}\backslash\\{[0,0]\\}$, let us define
$\Psi_{[u,v]}:\mathbb{R}^{+}\rightarrow\mathbb{R}$ by

$\displaystyle\Psi_{[u,v]}(t):=J[t\star u,t\star v]=$
$\displaystyle\frac{1}{2}\left[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\right]t^{2}-\frac{\mu_{1}}{p}\|u\|_{p}^{p}\,t^{\frac{(p-2)N}{2}}$
$\displaystyle-\frac{\mu_{2}}{q}\|v\|_{q}^{q}\,t^{\frac{(q-2)N}{2}}-\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}\,t^{\frac{(r_{1}+r_{2}-2)N}{2}}.$

A direct calculation shows that

$\displaystyle\Psi^{\prime}_{[u,v]}(t)=$ $\displaystyle\frac{d}{dt}J[t\star
u,t\star v]$ $\displaystyle=$ $\displaystyle\left[\|\nabla
u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\right]t-\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}\,t^{\frac{(p-2)N-2}{2}}$
$\displaystyle-\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}\,t^{\frac{(q-2)N-2}{2}}-\frac{(r_{1}+r_{2}-2)N}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}\,t^{\frac{(r_{1}+r_{2}-2)N-2}{2}}.$
(4.2)

We remark that $P[u,v]=\Psi^{\prime}_{[u,v]}(1)$ and $P[t\star u,t\star
v]=t\Psi^{\prime}_{[u,v]}(t)$.

######  Lemma 4.2.

Let $[u,v]\in\mathcal{H}\backslash\\{[0,0]\\}$. Then: $t\in\mathbb{R}^{+}$ is
a critical point of $\Psi_{[u,v]}(t)$ if and only if $[t\star u,t\star
v]\in\mathcal{P}$.

###### Proof.

By the fact $P[t\star u,t\star v]=t\Psi^{\prime}_{[u,v]}(t)$ and $t\neq 0$, we
see that

$[t\star u,t\star v]\in\mathcal{P}\Leftrightarrow P[t\star u,t\star
v]=0\Leftrightarrow\Psi^{\prime}_{[u,v]}(t)=0.$

∎

######  Corollary 4.3.

Assume that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. Then for any
$[0,0]\neq[u,v]\in D_{a}\times D_{b}$, there exists a unique $t=t_{[u,v]}>0$
such that $[t\star u,t\star v]\in\mathcal{P}_{(a,b)}$. Furthermore,
$t_{[u,v]}<(resp.\leavevmode\nobreak\ =,>)1$ if and only if
$P[u,v]<(resp.\leavevmode\nobreak\ =,>)0$.

###### Proof.

Noting that the fiber map (3.1) preserves the $L^{2}$-norm, we have that
$[t\star u,t\star v]\in D_{a}\times D_{b}$ for all $t>0$. Then by Lemma 4.2,
$[t\star u,t\star v]\in\mathcal{P}_{(a,b)}$ if and only if
$\Psi^{\prime}_{[u,v]}(t)=0$. Now by (4) and
$2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$, we obtain the uniqueness of
$t=t_{[u,v]}$. Furthermore, $t_{[u,v]}$ attains the maximum of
$\Psi_{[u,v]}(t)$ in $t\in\mathbb{R}^{+}$ and

$\Psi^{\prime}_{[u,v]}(t)>0\;\hbox{for}\leavevmode\nobreak\
0<t<t_{[u,v]}\;\hbox{while}\leavevmode\nobreak\
\Psi^{\prime}_{[u,v]}(t)<0\;\hbox{for}\leavevmode\nobreak\ t>t_{[u,v]}.$

So combining with $P[u,v]=\Psi^{\prime}_{[u,v]}(1)$, we obtain that

$P[u,v]<(resp.\leavevmode\nobreak\
=,>)0\Leftrightarrow\Psi^{\prime}_{[u,v]}(1)<(resp.\leavevmode\nobreak\
=,>)0\Leftrightarrow t_{[u,v]}<(resp.\leavevmode\nobreak\ =,>)1.$

∎

######  Lemma 4.4.

Suppose that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. There exists some
$C_{0}>0$ depending only on $p,q,r_{1},r_{2},N$, such that

$J[u,v]\geq C_{0}\Big{[}\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\Big{]},\quad\mbox{for all }[u,v]\leavevmode\nobreak\
\hbox{satisfying}\leavevmode\nobreak\ P[u,v]=0.$

###### Proof.

For $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$, set

$\tau:=\max\left\\{\frac{2}{(p-2)N},\frac{2}{(q-2)N},\frac{2}{(r_{1}+r_{2}-2)N}\right\\},$

then one can see that $0<\tau<\frac{1}{2}$. By the definition of $\tau$,

$\displaystyle\frac{\mu_{1}}{p}\|u\|_{p}^{p}+\frac{\mu_{2}}{q}\|v\|_{q}^{q}+\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\tau\left\\{\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}+\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}+\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\right\\}$

So for any $[u,v]$ with $P[u,v]=0$, we have that

$\frac{\mu_{1}}{p}\mu_{1}\|u\|_{p}^{p}+\frac{\mu_{2}}{q}\mu_{2}\|v\|_{q}^{q}+\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\leq\tau[\|\nabla
u\|_{2}^{2}+\|\nabla v\|_{2}^{2}].$

Hence, we can take $C_{0}:=\frac{1}{2}-\tau>0$ such that

$\displaystyle J[u,v]=$ $\displaystyle\frac{1}{2}[\|\nabla
u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]-\left[\frac{\mu_{1}}{p}\|u\|_{p}^{p}+\frac{\mu_{2}}{q}\|v\|_{q}^{q}+\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\right]$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]-\tau[\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}]$ $\displaystyle=$
$\displaystyle C_{0}[\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}].$

∎

######  Corollary 4.5.

Suppose that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. For any $a>0,b>0$,
$J\big{|}_{\mathcal{P}_{(a,b)}}$ is coercive, i.e.,

$\lim_{\stackrel{{\scriptstyle[u,v]\in\mathcal{P}_{(a,b)}}}{{\|\nabla
u\|_{2}^{2}+\|\nabla v\|_{2}^{2}\rightarrow+\infty}}}J[u,v]=+\infty.$

###### Proof.

It follows by Lemma 4.4. ∎

######  Lemma 4.6.

Suppose that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. For any given $a\geq
0,b\geq 0$ with $(a,b)\neq(0,0)$, there exists some $\delta_{(a,b)}>0$ such
that

$\inf_{(u,v)\in\mathcal{P}_{(a,b)}}\Big{[}\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\Big{]}\geq\delta_{(a,b)}.$

###### Proof.

Recalling the well known Gagliardo-Nirenberg inequality,

$\|u\|_{p}\leq C_{N,p}\|\nabla
u\|_{2}^{\frac{N(p-2)}{2p}}\|u\|_{2}^{1-\frac{N(p-2)}{2p}},\quad\mbox{for all
}u\in H^{1}(\mathbb{R}^{N})\leavevmode\nobreak\ \hbox{and all
}\leavevmode\nobreak\ \leavevmode\nobreak\ 2<p<2^{*},$

for any $[u,v]\in\mathcal{P}$, $P[u,v]=0$ implies that at least one of the
following holds.

  * (i)

$\displaystyle\frac{1}{3}[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]\leq\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}$,

  * (ii)

$\displaystyle\frac{1}{3}[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]\leq\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}$,

  * (iii)

$\displaystyle\frac{1}{3}[\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]\leq\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x$.

If $(i)$ holds, we have that

$\displaystyle\frac{1}{3}[\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}]\leq$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}C_{N,p}^{p}\|\nabla
u\|_{2}^{\frac{N(p-2)}{2}}\|u\|_{2}^{\frac{2N-(N-2)p}{4}}$ $\displaystyle\leq$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}C_{N,p}^{p}\left(\|\nabla
u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\right)^{\frac{N(p-2)}{4}}\left(\|u\|_{2}^{2}+\|v\|_{2}^{2}\right)^{\frac{2N-(N-2)p}{4}}.$

By $2+\frac{4}{N}<p<2^{*}$, we have that $\frac{N(p-2)}{4}>1$ and
$\frac{2N-(N-2)p}{4}>0$. Hence,

$\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\geq\left(\frac{3(p-2)N}{2p}\mu_{1}C_{N,p}^{p}\right)^{-\frac{4}{N(p-2)-4}}\left(\|u\|_{2}^{2}+\|v\|_{2}^{2}\right)^{-\frac{2N-(N-2)p}{N(p-2)-4}}.$
(4.3)

Similarly, if $(ii)$ holds, we have

$\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\geq\left(\frac{3(q-2)N}{2q}\mu_{2}C_{N,q}^{q}\right)^{-\frac{4}{N(q-2)-4}}\left(\|u\|_{2}^{2}+\|v\|_{2}^{2}\right)^{-\frac{2N-(N-2)q}{N(q-2)-4}}.$
(4.4)

If $(iii)$ holds, by Hölder inequality, a similar argument shows that

$\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}\geq\left(\frac{3(r_{1}+r_{2}-2)N}{2}\beta
C_{N,r_{1}+r_{2}}^{r_{1}+r_{2}}\right)^{-\frac{4}{N(r_{1}+r_{2}-2)-4}}\left(\|u\|_{2}^{2}+\|v\|_{2}^{2}\right)^{-\frac{2N-(N-2)(r_{1}+r_{2})}{N(r_{1}+r_{2}-2)-4}}.$
(4.5)

For $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$, by (4.3)-(4.5), we can find some
$\delta_{(a,b)}>0$ such that

$\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}\geq\delta_{(a,b)},\quad\mbox{for
all }[u,v]\in\mathcal{P}_{(a,b)}.$

∎

######  Corollary 4.7.

Assume that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. For any
$(a,b)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$,

$C_{(a,b)}=\inf_{[0,0]\neq[u,v]\in D_{a}\times D_{b}}\max_{t>0}J[t\star
u,t\star v]>0.$

###### Proof.

For any $[u,v]\in\mathcal{P}_{(a,b)}$, by Lemma 4.4 and Lemma 4.6, we have
that

$J[u,v]\geq C_{0}[\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}]\geq
C_{0}\delta_{(a,b)}>0.$

Hence, $C_{(a,b)}$ is well defined and $C_{(a,b)}\geq C_{0}\delta_{(a,b)}>0$.
Furthermore, by Corollary 4.3, one can see that

$\inf_{\mathcal{P}_{(a,b)}}J[u,v]=\inf_{[0,0]\neq[u,v]\in D_{a}\times
D_{b}}\max_{t>0}J[t\star u,t\star v].$

∎

We end this section by showing that any critical point of $J$ restricted to
$\mathcal{P}_{(a,b)}$ is a critical point of $J$ restricted to $D_{a}\times
D_{b}$. This implies, in particular, that any minimum of $C_{(a,b)}$ is a
ground state to Problem (1.4).

######  Lemma 4.8.

For any critical point of $J\big{|}_{\mathcal{P}_{(a,b)}}$, if
$\Psi^{\prime\prime}_{[u,v]}(1)\neq 0$, then there exists some
$\lambda_{1},\lambda_{2}\in\mathbb{R}$ such that

$J^{\prime}[u,v]+\lambda_{1}[u,0]+\lambda_{2}[0,v]=0.$

###### Proof.

Firstly, it is easy to see that there exists some
$\lambda_{1},\lambda_{2},\mu\in\mathbb{R}$ such that

$J^{\prime}[u,v]+\lambda_{1}[u,0]+\lambda_{2}[0,v]+\mu
P^{\prime}[u,v]=0\leavevmode\nobreak\ \hbox{in $\mathcal{H}^{-1}$, the dual
space of}\leavevmode\nobreak\ \mathcal{H}.$ (4.6)

Hence, we only need to prove that $\mu=0$. The functional associated to this
equation is given by

$\Phi[u,v]:=J[u,v]+\frac{1}{2}\lambda_{1}\|u\|_{2}^{2}+\frac{1}{2}\lambda_{2}\|v\|_{2}^{2}+\mu
P[u,v].$

The fact that $[u,v]$ solves (4.6) implies that $t=1$ is a critical point of
$\tilde{\Psi}_{[u,v]}:\mathbb{R}^{+}\rightarrow\mathbb{R}$ given by

$\displaystyle\tilde{\Psi}_{[u,v]}(t):=\Phi[t\star u,t\star v]=$
$\displaystyle J[t\star u,t\star
v]+\frac{\lambda_{1}}{2}\|u\|_{2}^{2}+\frac{\lambda_{2}}{2}\|v\|_{2}^{2}+\mu
P[t\star u,t\star v]$ $\displaystyle=$
$\displaystyle\Psi_{[u,v]}(t)+\frac{\lambda_{1}}{2}\|u\|_{2}^{2}+\frac{\lambda_{2}}{2}\|v\|_{2}^{2}+\mu
t\Psi^{\prime}_{[u,v]}(t).$

We have

$\tilde{\Psi}^{\prime}_{[u,v]}(t)=\frac{d}{dt}\Phi[t\star u,t\star
v]=(1+\mu)\Psi^{\prime}_{[u,v]}(t)+\mu\Psi^{\prime\prime}_{[u,v]}(t).$

Substituting $t=1$ and $\Psi^{\prime}_{[u,v]}(1)=P[u,v]=0$, we obtain that
$\mu\Psi^{\prime\prime}_{[u,v]}(1)=0$. Then it follows $\mu=0$ by
$\Psi^{\prime\prime}_{[u,v]}(1)\neq 0$. ∎

######  Lemma 4.9.

Suppose that $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. Then for any
$[u,v]\in\mathcal{P}$, we have $\Psi^{\prime\prime}_{[u,v]}(1)<0$.

###### Proof.

By a direct computation, we have that

$\displaystyle\Psi^{\prime\prime}_{[u,v]}(t)=$ $\displaystyle[\|\nabla
u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]-\frac{(p-2)N}{2p}\frac{(p-2)N-2}{2}\mu_{1}\|u\|_{p}^{p}\,t^{\frac{(p-2)N-4}{2}}$
$\displaystyle-\frac{(q-2)N}{2q}\frac{(q-2)N-2}{2}\mu_{2}\|v\|_{q}^{q}\,t^{\frac{(q-2)N-4}{2}}$
$\displaystyle-\frac{(r_{1}+r_{2}-2)N}{2}\frac{(r_{1}+r_{2}-2)N-2}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}t^{\frac{(r_{1}+r_{2}-2)N-4}{2}}.$

Hence,

$\displaystyle\Psi^{\prime\prime}_{[u,v]}(1)=$ $\displaystyle[\|\nabla
u\|_{2}^{2}+\|\nabla
v\|_{2}^{2}]-\frac{(p-2)N}{2p}\frac{(p-2)N-2}{2}\mu_{1}\|u\|_{p}^{p}$
$\displaystyle-\frac{(q-2)N}{2q}\frac{(q-2)N-2}{2}\mu_{2}\|v\|_{q}^{q}$
$\displaystyle-\frac{(r_{1}+r_{2}-2)N}{2}\frac{(r_{1}+r_{2}-2)N-2}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x.$

On the other hand, for any $[u,v]\in\mathcal{P}$, we have

$\displaystyle\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}=$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}+\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x.$

Hence,

$\displaystyle\Psi^{\prime\prime}_{[u,v]}(1)=$
$\displaystyle-\left[(p-2)N-4\right]\frac{(p-2)N}{4p}\mu_{1}\|u\|_{p}^{p}-\left[(q-2)N-4\right]\frac{(q-2)N}{4q}\mu_{2}\|v\|_{q}^{q}$
$\displaystyle-\left[(r_{1}+r_{2}-2)N-4\right]\frac{(r_{1}+r_{2}-2)N}{4}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x.$

So by $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$, and $[u,v]\neq[0,0]$, we see that
$\Psi^{\prime\prime}_{[u,v]}(1)<0$. ∎

##  5\. Rearrangement

For any $[u,v]\in\mathcal{P}_{(a,b)}$, let $u^{*},v^{*}$ be the Schwartz
symmetrization of $u$ and $v$. We refer to [14] for a definition and the main
properties of Schwartz symmetrization. We have the following result.

######  Lemma 5.1.

Let $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$. For any
$[u,v]\in\mathcal{P}_{(a,b)}$, there exists a unique
$t=t_{[u^{*},v^{*}]}\in(0,1]$ such that $[t\star u^{*},t\star
v^{*}]\in\mathcal{P}_{(a,b)}$ and we have $J[t\star u^{*},t\star v^{*}]\leq
J[u,v].$

###### Proof.

For any $[u,v]\in\mathcal{P}_{(a,b)}$, we have $[u,v]\neq[0,0]$. So by
$\|u^{*}\|_{2}^{2}=\|u\|_{2}^{2}$ and $\|v^{*}\|_{2}^{2}=\|v\|_{2}^{2}$, we
see that $[u^{*},v^{*}]\in(D_{a}\times D_{b})\backslash\\{[0,0]\\}$. Then by
Corollary 4.3, there exists a unique $t=t_{[u^{*},v^{*}]}>0$ such that
$[t\star u^{*},t\star v^{*}]\in\mathcal{P}_{(a,b)}$. In particular, by the
properties of rearrangement, we have

$\begin{cases}\|\nabla u^{*}\|_{2}^{2}\leq\|\nabla u\|_{2}^{2},\|\nabla
v^{*}\|_{2}^{2}\leq\|\nabla v\|_{2}^{2},\\\
\|u^{*}\|_{p}^{p}=\|u\|_{p}^{p},\|v^{*}\|_{q}^{q}=\|v\|_{q}^{q},\\\
\int_{\mathbb{R}^{N}}|u^{*}|^{r_{1}}|v^{*}|^{r_{2}}\mathrm{d}x\geq\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x.\end{cases}$

This implies that $P[u^{*},v^{*}]\leq P[u,v]=0$. Hence, by Corollary 4.3, we
obtain $t_{[u^{*},v^{*}]}\leq 1$. We remark that for any $[u,v]\neq[0,0]$, we
have that $J[u^{*},v^{*}]\leq J[u,v]$. Combined with the fact that

$s\star u^{*}=(s\star u)^{*},\quad\mbox{for all }s\in\mathbb{R}^{+},$

we finally obtain that

$\displaystyle\max_{s>0}J[s\star u^{*},s\star v^{*}]=J[t\star u^{*},t\star
v^{*}]=$ $\displaystyle J[(t\star u)^{*},(t\star v)^{*}]\leq J[t\star u,t\star
v]$ $\displaystyle\leq$ $\displaystyle\max_{s>0}J[s\star u,s\star v]=J[u,v].$

∎

######  Remark 5.2.

Lemma 5.1 implies, in particular, that if $C_{(a,b)}$ is achieved then it is
achieved by a couple $[u,v]\in\mathcal{H}$ of Schwartz symmetric functions.

##  6\. Existence of a special Palais-Smale sequence at the level $C_{(a,b)}$

In this section we prove the existence of a particular Palais-Smale sequence
for $J$ restricted to $D_{a}\times D_{b}$ at the level $C_{(a,b)}$.

Let us define the class of paths

$\Gamma_{(a,b)}=\\{g\in C([0,1],D_{a}\times D_{b}):g(0)\in
L_{(a,b)}\,\,\mbox{and}\,\,P(g(1))<0\\}$

where

$L_{(a,b)}:=\\{[u,v]\in\mathcal{H}:||\nabla u||_{2}^{2}+||\nabla
u||_{2}^{2}\leq\frac{1}{2}\delta_{(a,b)}\\}.$

Note that $\Gamma_{(a,b)}\neq\emptyset.$ Indeed $L_{(a,b)}\neq\emptyset$ and
from Corollary 4.3 we known that the set where $P<0$ is non void.

Next we define

$\gamma_{(a,b)}=\inf_{g\in\Gamma_{(a,b)}}\max_{t\in[0,1]}J(g(t)).$

We claim that

$\gamma_{(a,b)}=C_{(a,b)}.$

Indeed, combining Corollary 4.3 and Lemma 4.6 we see that $P>0$ on $L_{(a,b)}$
and thus, by continuity, for any $g\in\Gamma_{(a,b)}$ there exists a
$t\in[0,1]$ such that $P(g(t))=0$. It implies that $\gamma_{(a,b)}\geq
C_{(a,b)}$. The reverse inequality $\gamma_{(a,b)}\leq C_{(a,b)}$ follows from
Corollary 4.3.

As a consequence of the equality $\gamma_{(a,b)}=C_{(a,b)}$ we deduce that

$\inf_{u\in\mathcal{P}_{(a,b)}}J(u)\geq\gamma_{(a,b)}.$

At this point we observe that we are in the setting of [10, Theorem 4.1]. This
theorem says to us that, for any sequence of paths
$\\{g_{n}\\}\subset\Gamma_{(a,b)}$ such that, as $n\rightarrow\infty$,

$\max_{[0,1]}J(g_{n}(t))\rightarrow C_{(a,b)},$ (5.1)

there exists a sequence $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$ such that, as
$n\rightarrow\infty$,

  * (i)

$J[u_{n},v_{n}]\rightarrow C_{(a,b)}$,

  * (ii)

$J^{\prime}[u_{n},v_{n}]\Big{|}_{D_{a}\times D_{b}}\rightarrow 0$,

  * (iii)

$dist([u_{n},v_{n}],\mathcal{P}_{(a,b)})\rightarrow 0$,

  * (iv)

$dist([u_{n},v_{n}],g_{n}[0,1])\rightarrow 0.$

Let $[u_{n}^{*},v_{n}^{*}]\subset\mathcal{P}_{(a,b)}$ be a minimizing sequence
for $C_{(a,b)}$. We know by Lemma 5.1 that such a sequence exists. Now let the
sequence of paths $\\{g_{n}\\}\subset\Gamma_{(a,b)}$ be given by :

$g_{n}(t)=[(s_{1}^{n}+ts_{2}^{n})*u_{n}^{*},(s_{1}^{n}+ts_{2}^{n})*v_{n}^{*}]$

where $s_{1}^{n}>0$ is sufficiently small and $s_{2}^{n}>0$ sufficiently large
in order to insure that $g_{n}\in\Gamma_{(a,b)}$. We know from Corollary 4.7
that the sequence $\\{g_{n}\\}$ satisfy (5.1).

From (i)-(ii) we deduce that $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$ is a
Palais-Smale sequence for $J$ restricted to $D_{a}\times D_{b}$ at the level
$C_{(a,b)}$. The property (iii), combined with the fact that $J$ is coercive
on $\mathcal{P}_{(a,b)}$, see Corollary 4.5, and that $J^{\prime}$ and
$P^{\prime}$ take bounded sets into bounded set implies that
$\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$ is bounded and that
$P[u_{n},v_{n}]\rightarrow 0$. Now, by compactness, for each $n\in\mathbb{N}$,
there exists $s_{n}\in[s_{1}^{n},s_{1}^{n}+s_{2}^{n}]$ such that

$dist\Big{(}[u_{n},v_{n}],g_{n}([0,1])\Big{)}=\Big{|}\Big{|}[u_{n},v_{n}]-[s_{n}*u_{n}^{*},s_{n}*v_{n}^{*}]\Big{|}\Big{|}.$
(5.2)

Up to a subsequence we can assume that
$[s_{n}*u_{n}^{*},s_{n}*v_{n}^{*}]\rightharpoonup u$ weakly in $\mathcal{H}$
and $[s_{n}*u_{n}^{*},s_{n}*v_{n}^{*}]\rightarrow u$ strongly in
$L^{\eta}(\mathbb{R}^{N})\times L^{\eta}(\mathbb{R}^{N})$ for all
$2<\eta<2^{*}$. For $N\geq 2$ this strong convergence is a consequence of the
fact that the functions are radial. When $N=1$ it follows from the fact that
the functions are nonincreasing with respect to the origin, see [8,
Proposition 1.7.1].

Now, in view of (5.2), these convergence properties hold for the Palais-Smale
sequence $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}.$

Summarizing we have obtained a bounded sequence $\\{[u_{n},v_{n}]\\}\subset
D_{a}\times D_{b}$ such that

  * (i)

$J[u_{n},v_{n}]\rightarrow C_{(a,b)}$,

  * (ii)

$J^{\prime}[u_{n},v_{n}]+\lambda_{1,n}[u_{n},0]+\lambda_{2,n}[0,v_{n}]\to
0,\quad\mbox{in }\mathcal{H}^{-1}$ for some real sequences
$\\{\lambda_{1,n}\\}$ and $\\{\lambda_{2,n}\\}$,

  * (iii)

$P[u_{n},v_{n}]\rightarrow 0$.

  * (iv)

$[u_{n},v_{n}]\rightharpoonup[u,v]$ weakly in $\mathcal{H}$ and
$[u_{n},v_{n}]\rightarrow[u,v]$ strongly in $L^{\eta}(\mathbb{R}^{N})\times
L^{\eta}(\mathbb{R}^{N})$ for all $2<\eta<2^{*}$. In addition $u\geq 0$ and
$v\geq 0$.

##  7\. Estimation of $C_{(a,b)}$

In view of the definitions of $z_{p,\mu,a}$ by (2.4) and $\beta_{p,\mu,a,N,r}$
by (2.5), a direct calculation shows that

$\beta_{p,\mu,a,N,r}=\frac{1}{2}\lambda_{p,\mu,a}^{\frac{p-2-r}{p-2}}\mu^{\frac{r}{p-2}}\inf_{h\in
H^{1}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\int_{\mathbb{R}^{N}}|\nabla
h|^{2}\mathrm{d}x}{\int_{\mathbb{R}^{N}}U_{p}^{r}h^{2}\mathrm{d}x}.$

Substituting the value of $\lambda_{p,\mu,a}$ (see (2.3)), we obtain that

$\beta_{p,\mu,a,N,r}=\frac{1}{2}\mu^{\frac{rN-4}{N(p-2)-4}}\,\|U_{p}\|_{2}^{\frac{4(p-2-r)}{N(p-2)-4}}a^{-\frac{2(p-2-r)}{N(p-2)-4}}\inf_{h\in
H^{1}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\int_{\mathbb{R}^{N}}|\nabla
h|^{2}\mathrm{d}x}{\int_{\mathbb{R}^{N}}U_{p}^{r}h^{2}\mathrm{d}x}.$

######  Lemma 7.1.

For $N=1,2$, $p\in(2,+\infty),\mu>0,a>0,r>0$, it holds that
$\beta_{p,\mu,a,N,r}=0$.

###### Proof.

For the case of $N=1$, let $h(x)\in H^{1}(\mathbb{R})$ such that $h(x)\equiv
1$ for $|x|\leq 1$, $h(x)\equiv 0$ for $|x|\geq 2$ and $0\leq h(x)\leq 1$.
Define $\displaystyle h_{n}(x):=h\left(\frac{x}{n}\right)$. Then $\|\nabla
h_{n}\|_{2}^{2}=\frac{1}{n}\|\nabla h\|_{2}^{2}\rightarrow 0$ as
$n\rightarrow\infty$. While

$\int_{\mathbb{R}}U_{p}^{r}(x)h_{n}^{2}(x)\mathrm{d}x\geq\int_{|x|\leq
n}U_{p}^{r}(x)\mathrm{d}x\rightarrow\int_{\mathbb{R}}U_{p}^{r}(x)\mathrm{d}x>0.$

So

$\inf_{0\neq h\in H^{1}(\mathbb{R})}\frac{\|\nabla
h\|_{2}^{2}}{\int_{\mathbb{R}}U_{p}^{r}(x)h(x)^{2}\mathrm{d}x}=0$

and thus $\beta_{p,\mu,a,N,r}=0$ for $N=1$.

For $N=2$, we remark that the embedding $H^{1}(\mathbb{R}^{2})\hookrightarrow
L^{\infty}(\mathbb{R}^{2})$ fails, so there exists a sequence
$\\{\phi_{n}\\}\subset H^{1}(\mathbb{R}^{N})$ with $\|\phi_{n}\|_{H^{1}}=1$,
but $\|\phi_{n}\|_{\infty}\rightarrow\infty$ as $n\rightarrow\infty$. In
particular, by the property of rearrangement, without loss of generality, we
can assume that $\phi_{n}=\phi_{n}^{*}$. So we have that

$\phi_{n}(x)=\phi_{n}(|x|)\leavevmode\nobreak\ \hbox{and}\leavevmode\nobreak\
\phi_{n}(0)=\|\phi_{n}\|_{\infty}.$

Now, we set

$h_{n}(x)=\begin{cases}\phi_{n}(0)\quad&\hbox{if}\leavevmode\nobreak\ |x|\leq
n,\\\ \phi_{n}(|x|-n)\quad&\hbox{if}\leavevmode\nobreak\ |x|\geq
n.\end{cases}$

Then $\|\nabla
h_{n}\|_{2}^{2}=\|\nabla\phi_{n}\|_{2}^{2}\leq\|\phi_{n}\|_{H^{1}}^{2}=1$ and

$\displaystyle\int_{\mathbb{R}^{2}}U_{p}^{r}(x)h_{n}^{2}(x)\mathrm{d}x\geq\int_{|x|\leq
n}\phi_{n}^{2}(0)U_{p}^{r}(x)\mathrm{d}x=\|\phi_{n}\|_{\infty}^{2}\int_{|x|\leq
n}U_{p}^{r}(x)\mathrm{d}x\rightarrow\infty\leavevmode\nobreak\
\hbox{as}\leavevmode\nobreak\ n\rightarrow\infty.$

So we also have

$\inf_{0\neq h\in H^{1}(\mathbb{R}^{2})}\frac{\|\nabla
h\|_{2}^{2}}{\int_{\mathbb{R}^{2}}U_{p}^{r}(x)h^{2}(x)}\mathrm{d}x=0,$

and thus $\beta_{p,\mu,a,N,r}=0$ also holds for $N=2$. ∎

######  Lemma 7.2.

For $N\geq 3$, it holds that $\beta_{p,\mu,a,N,r}>0$. In particular, for
$r>\frac{4}{N}$ and fixed $2+\frac{4}{N}<p<2^{*},a>0$, we have

$\lim_{\mu\rightarrow\infty}\beta_{p,\mu,a,N,r}=+\infty.$ (6.1)

Furthermore, for $2+\frac{4}{N}<p<2^{*},\mu>0,r>1$ fixed, we have that

  * (i)

if $p<r+2<2^{*}$, then $\displaystyle\lim_{a\rightarrow
0}\beta_{p,\mu,a,N,r}=0$ and
$\displaystyle\lim_{a\rightarrow\infty}\beta_{p,\mu,a,N,r}=+\infty$.

  * (ii)

if $r+2=p$, then there exists $\tilde{C}_{p,\mu}<\tilde{D}_{p,\mu}$
independent of $a$, such that

$\tilde{C}_{p,\mu}\leq\beta_{p,\mu,a,N,r}\leq\tilde{D}_{p,\mu},\forall a>0.$

  * (iii)

if $2+r<p$, then $\displaystyle\lim_{a\rightarrow
0}\beta_{p,\mu,a,N,r}=+\infty$ and
$\displaystyle\lim_{a\rightarrow\infty}\beta_{p,\mu,a,N,r}=0$.

###### Proof.

Noting that $U_{p}\in L^{\infty}(\mathbb{R}^{N})$ and decays exponentially to
$0$ as $|x|\rightarrow+\infty$, by Hölder inequality, we have that

$\int_{\mathbb{R}^{N}}U_{p}^{r}|h|^{2}\mathrm{d}x\leq\|U_{p}\|_{\frac{2^{*}r}{2^{*}-2}}^{r}\|h\|_{2^{*}}^{2}.$

So by the critical Sobolev inequality, we obtain that

$\displaystyle\beta_{p,\mu,a,N,r}\geq$
$\displaystyle\frac{1}{2}\mu^{\frac{rN-4}{N(p-2)-4}}\,\|U_{p}\|_{2}^{\frac{4(p-2-r)}{N(p-2)-4}}a^{-\frac{2(p-2-r)}{N(p-2)-4}}\frac{\|\nabla
h\|_{2}^{2}}{\|U_{p}\|_{\frac{2^{*}r}{2^{*}-2}}^{r}\|h\|_{2^{*}}^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\mu^{\frac{rN-4}{N(p-2)-4}}\,\|U_{p}\|_{2}^{\frac{4(p-2-r)}{N(p-2)-4}}a^{-\frac{2(p-2-r)}{N(p-2)-4}}\|U_{p}\|_{\frac{2^{*}r}{2^{*}-2}}^{-r}S_{N},$
(6.2)

where $S_{N}>0$ is the sharp constant in the critical Sobolev inequality. So
it is easy to see that (6.1) holds.

On the other hand, by taking $h=U_{p}$, we obtain that

$\beta_{p,\mu,a,N,r}\leq\frac{1}{2}\mu^{\frac{rN-4}{N(p-2)-4}}\|U_{p}\|_{2}^{\frac{4(p-2-r)}{N(p-2)-4}}a^{-\frac{2(p-2-r)}{N(p-2)-4}}\frac{\|\nabla
U_{p}\|_{2}^{2}}{\|U_{p}\|_{r+2}^{r+2}}.$ (6.3)

Then by (7) and (6.3), it is easy to see that the conclusions (i)-(iii) above
hold. ∎

######  Lemma 7.3.

  * (i)

if $1<r_{1}<2$, then $C_{(a,b)}<m_{q,\mu_{2},b}$. If $r_{1}=2$, then
$C_{(a,b)}<m_{q,\mu_{2},b}$ provided $\beta>\beta_{q,\mu_{2},b,N,r_{2}}$.

  * (ii)

if $1<r_{2}<2$, then $C_{(a,b)}<m_{p,\mu_{1},a}$. If $r_{2}=2$, then
$C_{(a,b)}<m_{p,\mu_{1},a}$ provided $\beta>\beta_{p,\mu_{1},a,N,r_{1}}$.

###### Proof.

We only prove (ii), the proof of (i) being directly similar. We write
$z:=z_{p,\mu_{1},a}$ for simplicity. For any $h\in H^{1}(\mathbb{R}^{N})$ with
$\|h\|_{2}^{2}=1$, we have that $[z,sh]\in D_{a}\times D_{b}$ provided
$|s|<\sqrt{b}$. For any $s$, there exists a unique $t=t(s)>0$ such that
$[t\star z,t\star sh]\in\mathcal{P}_{(a,b)}$, where $t=t(s)$ is determined by

$\displaystyle\|\nabla z\|_{2}^{2}+s^{2}\|\nabla h\|_{2}^{2}=$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}\|z\|_{p}^{p}\,t^{\frac{(p-2)N-4}{2}}$
$\displaystyle+\frac{(q-2)N}{2q}\mu_{2}\|h\|_{q}^{q}\,s^{q}\,t^{\frac{(q-2)N-4}{2}}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}s^{r_{2}}\,t^{\frac{(r_{1}+r_{2}-2)N-4}{2}}.$

From the Implicit Function Theorem we know that $t(s)\in C^{1}$ locally around
$s=0$. Taking the derivative with respect to $s$ at both sides and we obtain
that

$\displaystyle t^{\prime}(s)=\frac{P_{h}(s)}{Q_{h}(s)},$

where

$\displaystyle P_{h}(s):=$ $\displaystyle 2\|\nabla
h\|_{2}^{2}s-\frac{(q-2)N}{2}\mu_{2}\,\|h\|_{q}^{q}\,|s|^{q-2}s\,t^{\frac{(q-2)N-4}{2}}$
$\displaystyle-\frac{(r_{1}+r_{2}-2)N}{2}\beta\,r_{2}\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}|s|^{r_{2}-2}s\,t^{\frac{(r_{1}+r_{2}-2)N-4}{2}}$

and

$\displaystyle Q_{h}(s):=$
$\displaystyle\frac{(p-2)N-4}{2}\frac{(p-2)N}{2p}\mu_{1}\,\|z\|_{p}^{p}\,t^{\frac{(p-2)N-6}{2}}$
$\displaystyle+\frac{(q-2)N-4}{2}\frac{(q-2)N}{2q}\mu_{2}\,\|h\|_{q}^{q}\,s^{q}\,t^{\frac{(q-2)N-6}{2}}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N-4}{2}\frac{(r_{1}+r_{2}-2)N}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}s^{r_{2}}\,t^{\frac{(r_{1}+r_{2}-2)N-6}{2}}.$

Case of $1<r_{2}<2$: In such a case, for $|s|$ small,

$P_{h}(s)=-\frac{(r_{1}+r_{2}-2)N}{2}\beta
r_{2}\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}|s|^{r_{2}-2}s(1+o(1))$

and

$Q_{h}(s)=\frac{(p-2)N-4}{2}\,\frac{(p-2)N}{2p}\mu_{1}\|z\|_{p}^{p}(1+o(1)).$

So

$t^{\prime}(s)=-M_{h}r_{2}|s|^{r_{2}-2}s(1+o(1)),$

where

$M_{h}:=\frac{2(r_{1}+r_{2}-2)N}{(p-2)N-4}\,\frac{\beta\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x}{\|\nabla
z\|_{2}^{2}}.$

Then

$t(s)=1-M_{h}|s|^{r_{2}}(1+o(1))$

and for any $\tau>0$,

$t(s)^{\tau}=1-\tau M_{h}|s|^{r_{2}}(1+o(1)).$

Noting that for any $[\phi,\psi]\in\mathcal{P}$, it holds that

$\displaystyle J[\phi,\psi]=$
$\displaystyle\frac{(p-2)N-4}{4}\,\frac{\mu_{1}}{p}\|\phi\|_{p}^{p}+\frac{(q-2)N-4}{4}\,\frac{\mu_{2}}{q}\|\psi\|_{q}^{q}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N-4}{4}\beta\int_{\mathbb{R}^{N}}|\phi|^{r_{1}}|\psi|^{r_{2}}\mathrm{d}x.$

Then we have

$\displaystyle J[t(s)\star z,t(s)\star(sh)]-J[z,0]$
$\displaystyle=\frac{1}{2}\|\nabla
z\|_{2}^{2}\,(t^{2}(s)-1)-\frac{\mu_{1}}{p}\,\|z\|_{p}^{p}\,(t(s)^{\frac{(p-2)N}{2}}-1)$
$\displaystyle+\frac{1}{2}\|\nabla
h\|_{2}^{2}\,s^{2}\,t^{2}(s)-\frac{\mu_{2}}{q}\,\,\|h\|_{q}^{q}\,s^{q}\,t(s)^{\frac{(q-2)N}{2}}$
$\displaystyle-\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}|s|^{r_{2}}\,t(s)^{\frac{(r_{1}+r_{2}-2)N}{2}}$
$\displaystyle=$ $\displaystyle-\|\nabla
z\|_{2}^{2}\,M_{h}\,|s|^{r_{2}}o(1)-\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}|s|^{r_{2}}(1+o(1))$
$\displaystyle=$
$\displaystyle-\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{r_{2}}\mathrm{d}x\Big{)}|s|^{r_{2}}(1+o(1)).$

Hence, if $1<r_{2}<2$, by taking $s$ close to $0$, we obtain that

$C_{(a,b)}\leq J[t(s)\star z,t(s)\star(sh)]<J[z,0]=m_{p,\mu_{1},a}.$

Case of $r_{2}=2$: In such a case, we have, for $s>0$ small,

$P_{h}(s)=\left[2\|\nabla h\|_{2}^{2}-N\beta
r_{1}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{2}\mathrm{d}x\right]s(1+o(1))$

and

$t^{\prime}(s)=\bar{M}_{h}s(1+o(1))$

where

$\bar{M}_{h}:=\frac{2\|\nabla h\|_{2}^{2}-N\beta
r_{1}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{2}\mathrm{d}x}{\frac{(p-2)N-4}{2}\|\nabla
z\|_{2}^{2}}.$

Then

$t(s)=1+\frac{1}{2}\bar{M}_{h}\,s^{2}(1+o(1))$

and for any $\tau>0$,

$t(s)^{\tau}=1+\frac{\tau}{2}\bar{M}_{h}\,s^{2}(1+o(1)).$

So by a similar argument as in the Case of $1<r_{2}<2$, we obtain that

$\displaystyle J[t(s)\star z,t(s)\star(sh)]-J[z,0]=$
$\displaystyle\frac{1}{2}\|\nabla
z\|_{2}^{2}\,(t(s)^{2}-1)-\frac{1}{p}\,\mu_{1}\|z\|_{p}^{p}(t(s)^{\frac{(p-2)N}{2}}-1)$
$\displaystyle+\frac{1}{2}\|\nabla
h\|_{2}^{2}\,s^{2}\,t(s)^{2}-\frac{1}{q}\mu_{2}\,\|h\|_{q}^{q}\,s^{q}\,t(s)^{\frac{(q-2)N}{2}}$
$\displaystyle-\beta\Big{(}\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{2}\mathrm{d}x\Big{)}|s|^{2}t(s)^{\frac{r_{1}N}{2}}$
$\displaystyle=$ $\displaystyle\left[\frac{1}{2}\|\nabla
h\|_{2}^{2}-\beta\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{2}\mathrm{d}x\right]s^{2}(1+o(1)).$

Then for any $\beta>\beta_{p,\mu_{1},a,N,r_{1}}$, there exists some $h\in
H^{1}(\mathbb{R}^{N})$ such that

$\frac{1}{2}\|\nabla
h\|_{2}^{2}-\beta\int_{\mathbb{R}^{N}}|z|^{r_{1}}|h|^{2}\mathrm{d}x<0,$

and for such a $h$ and $|s|$ small, we have $J[t(s)\star
z,t(s)\star(sh)]-J[z,0]<0$. Hence,

$C_{(a,b)}<m_{p,\mu_{1},a}.$

∎

##  8\. Convergence of our Palais-Smale sequence and Proof of Theorem 2.2

Theorem 2.2 will be a direct consequence of the following result.

######  Theorem 8.1.

Let $1\leq N\leq 4$, $2+\frac{4}{N}<p,q,r_{1}+r_{2}<2^{*}$ and
$\mu_{1},\mu_{2},\beta,a,b\in\mathbb{R}^{+}$. Let $\\{[u_{n},v_{n}]\\}\subset
D_{a}\times D_{b}$ be a bounded sequence such that

  * (i)

$J[u_{n},v_{n}]\rightarrow C_{(a,b)}$,

  * (ii)

$J^{\prime}[u_{n},v_{n}]+\lambda_{1,n}[u_{n},0]+\lambda_{2,n}[0,v_{n}]\to
0,\quad\mbox{in }\mathcal{H}^{-1}$ for some real sequences
$\\{\lambda_{1,n}\\}$ and $\\{\lambda_{2,n}\\}$,

  * (iii)

$P[u_{n},v_{n}]\rightarrow 0$,

  * (iv)

$[u_{n},v_{n}]\rightharpoonup[u,v]$ weakly in $\mathcal{H}$ and
$[u_{n},v_{n}]\rightarrow[u,v]$ strongly in $L^{\eta}(\mathbb{R}^{N})\times
L^{\eta}(\mathbb{R}^{N})$ for all $2<\eta<2^{*}$. In addition $u\geq 0$ and
$v\geq 0$.

Assume that

$C_{(a,b)}<\min\\{m_{p,\mu_{1},a},m_{q,\mu_{2},b}\\}.$ (7.1)

Then, up to a subsequence, $[u_{n},v_{n}]\rightarrow[u,v]$ in $\mathcal{H}$
and $[u,v]\in S_{a}\times S_{b}$.

###### Proof.

Without loss of generality, we may assume that $[u_{n},v_{n}]\neq[0,0]$ for
all $n\in\mathbb{N}$ since $J[u_{n},v_{n}]\rightarrow C_{a,b}>0=J[0,0]$, see
Corollary 4.7. First note that (ii) can be rewrite as : there exists sequences
$\\{\lambda_{1,n}\\}$ and $\\{\lambda_{2,n}\\}$ such that

$\begin{cases}-\Delta u_{n}+\lambda_{1,n}u_{n}-\mu_{1}u_{n}^{p-1}-\beta
r_{1}u_{n}^{r_{1}-1}v_{n}^{r_{2}}=o(1),\\\ -\Delta
v_{n}+\lambda_{2,n}v_{n}-\mu_{2}v_{n}^{q-1}-\beta
r_{2}u_{n}^{r_{1}}v_{n}^{r_{2}-1}=o(1).\end{cases}$

From the boundedness of $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$, it follows
that $\\{\lambda_{1,n}\\}$ and $\\{\lambda_{2,n}\\}$ are also bounded and we
can assume that $\lambda_{1,n}\rightarrow\lambda_{1}$ and
$\lambda_{2,n}\rightarrow\lambda_{2}$ for some $\lambda_{1},\lambda_{2}$.

We claim that

$u\neq 0,v\neq 0.$ (7.2)

If not, without loss of generality, we may assume that $u=0$. Then we have
that $v\neq 0$. If not, from $P[u_{n},v_{n}]\to 0$ we get that

$\displaystyle\|\nabla u_{n}\|_{2}^{2}+\|\nabla v_{n}\|_{2}^{2}=$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}\|u_{n}\|_{p}^{p}+\frac{(q-2)N}{2q}\mu_{2}\|v_{n}\|_{q}^{q}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u_{n}|^{r_{1}}|v_{n}|^{r_{2}}\mathrm{d}x+o(1),$
$\displaystyle=$ $\displaystyle o(1),$

where we have used that

$\|u_{n}\|_{p}^{p}=o(1),\|v_{n}\|_{q}^{q}=o(1)\mbox{ and
}\int_{\mathbb{R}^{N}}u_{n}^{r_{1}}v_{n}^{r_{2}}\mathrm{d}x=o(1),$

by the Hölder inequality and Gagliardo-Nirenberg inequality. This contradicts
Lemma 4.6. We deduce that

$0<\|v\|_{2}^{2}\leq\liminf_{n\rightarrow\infty}\|v_{n}\|_{2}^{2}.$

Thus one can see that $v$ is a non-trivial (non-negative) solution to

$-\Delta v+\lambda_{2}v=\mu_{2}v^{q-1}\;\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\,v\in H^{1}(\mathbb{R}^{N}).$ (7.3)

Recalling the Pohozaev identity

$\|\nabla v\|_{2}^{2}=\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q},$

we deduce, from (7.3), that

$\lambda_{2}\|v\|_{2}^{2}=\frac{2q-(q-2)q}{2q}\mu_{2}\|v\|_{q}^{q}>0.$

Hence, $\lambda_{2}>0$. So by

$\displaystyle\|\nabla v_{n}\|_{2}^{2}+\lambda_{2,n}\|v_{n}\|_{2}^{2}=$
$\displaystyle\mu_{2}\|v_{n}\|_{q}^{q}+\beta
r_{2}\int_{\mathbb{R}^{N}}u_{n}^{r_{1}}v_{n}^{r_{2}}\mathrm{d}x+o(1)$
$\displaystyle=$ $\displaystyle\mu_{2}\|v\|_{q}^{q}+o(1)$ $\displaystyle=$
$\displaystyle\|\nabla v\|_{2}^{2}+\lambda_{2}\|v\|_{2}^{2}+o(1),$

we have that $v_{n}\rightarrow v$ in $H^{1}(\mathbb{R}^{N})$. Now using that
$P[u_{n},v_{n}]\to 0,$ and $u_{n}\rightarrow 0$ in $L^{\eta}(\mathbb{R}^{N})$
for all $2<\eta<2^{*}$, it is easy to prove that $u_{n}\rightarrow 0$ in
$D_{0}^{1,2}(\mathbb{R}^{N})$. So

$J[0,v]=\lim_{n\rightarrow\infty}J[u_{n},v_{n}]=C_{(a,b)}.$

Put $\delta:=\|v\|_{2}^{2}$, then $\delta\in(0,b]$. By Lemma 3.1, one has that
$v=z_{q,\mu_{2},\delta}$ and $C_{(a,b)}=J[0,v]=m_{q,\mu_{2},\delta}$. However,
by Lemma 3.2-(i), and (7.1) we have

$m_{q,\mu_{2},\delta}\geq m_{q,\mu_{2},b}>C_{(a,b)},$

a contradiction. This ends the proof of (7.2) and thus

$0<\|u\|_{2}^{2}\leq\liminf_{n\rightarrow\infty}\|u_{n}\|_{2}^{2}\quad\mbox{and}\quad
0<\|v\|_{2}^{2}\leq\liminf_{n\rightarrow\infty}\|v_{n}\|_{2}^{2}.$

Now, from the convergence properties (iv), we deduce that $[u,v]$ is a non-
trivial, solution to (4.1). In particular it holds that $P[u,v]=0$ and by

$\displaystyle\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}\leq o(1)+\|\nabla
u_{n}\|_{2}^{2}+\|\nabla v_{n}\|_{2}^{2}$ $\displaystyle=$ $\displaystyle
o(1)+\frac{(p-2)N}{2p}\mu_{1}\|u_{n}\|_{p}^{p}+\frac{(q-2)N}{2q}\mu_{2}\|v_{n}\|_{q}^{q}+\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u_{n}|^{r_{1}}|v_{n}|^{r_{2}}\mathrm{d}x$
$\displaystyle=$ $\displaystyle
o(1)+\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}+\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}+\frac{(r_{1}+r_{2}-2)N}{2}\beta\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x$
$\displaystyle=$ $\displaystyle o(1)+\|\nabla u\|_{2}^{2}+\|\nabla
v\|_{2}^{2},$

we obtain that $u_{n}\rightarrow u,v_{n}\rightarrow v$ in
$D_{0}^{1,2}(\mathbb{R}^{N})$. Hence, $\displaystyle
J[u,v]=\lim_{n\rightarrow\infty}J[u_{n},v_{n}]=C_{(a,b)}.$

Recalling (4.1) and $1\leq N\leq 4$, we conclude, using [12, Lemma A.2], that
$\lambda_{1}>0,\lambda_{2}>0$. Then by

$\displaystyle\|\nabla u_{n}\|_{2}^{2}+\lambda_{1,n}\|u_{n}\|_{2}^{2}=$
$\displaystyle\mu_{1}\|u_{n}\|_{p}^{p}+\beta
r_{1}\int_{\mathbb{R}^{N}}u_{n}^{r_{1}}v_{n}^{r_{2}}\mathrm{d}x+o(1)$
$\displaystyle=$ $\displaystyle\mu_{1}\|u\|_{p}^{p}+\beta
r_{1}\int_{\mathbb{R}^{N}}u^{r_{1}}v^{r_{2}}\mathrm{d}x+o(1)$ $\displaystyle=$
$\displaystyle\|\nabla u\|_{2}^{2}+\lambda_{1}\|u\|_{2}^{2}+o(1),$

we obtain that $u_{n}\rightarrow u$ in $H^{1}(\mathbb{R}^{N})$. Similarly, we
can prove that $v_{n}\rightarrow v$ in $H^{1}(\mathbb{R}^{N})$. To obtain that
$[u,v]\in S_{a}\times S_{b}$ and completed the proof we shall use Lemma 8.2
below. ∎

######  Lemma 8.2.

Let $[u,v],\lambda_{1},\lambda_{2}$ be given by Theorem 8.1. Since
$\lambda_{1}>0$ we have that $u\in S_{a}$. Similarly, $\lambda_{2}>0$ imply
that $v\in S_{b}$.

###### Proof.

Suppose that $\lambda_{1}>0$, we shall prove that $u\in S_{a}$. If not,
$\delta:=\|u\|_{2}^{2}\in(0,a)$. Then for $s>0$ small enough, we still have
that $[(1+s)u,v]\in(D_{a}\times D_{b})\backslash\\{[0,0]\\}$. For any given
$s$, by Corollary 4.3, there exists a unique $t=t(s)>0$ such that
$[t\star(1+s)u,t\star v]\in\mathcal{P}_{(a,b)}$. Precisely, $t=t(s)$ is
determined by

$\displaystyle(1+s)^{2}\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}=$
$\displaystyle\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}\,t^{\frac{(p-2)N-4}{2}}(1+s)^{p}$
$\displaystyle+\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}\,t^{\frac{(q-2)N-4}{2}}$
$\displaystyle+\frac{(r_{1}+r_{2}-2)N}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}t^{\frac{(r_{1}+r_{2}-2)N-4}{2}}(1+s)^{r_{1}}.$

From the Implicit Function Theorem we have that $t(s)\in C^{1}$. Then, since

$\displaystyle J[t\star(1+s)u,t\star v]=$ $\displaystyle\frac{1}{2}[\|\nabla
u\|_{2}^{2}(1+s)^{2}+\|\nabla
v\|_{2}^{2}]t^{2}-\frac{\mu_{1}}{p}\|u\|_{p}^{p}(1+s)^{p}\,t^{\frac{(p-2)N}{2}}$
$\displaystyle-\frac{\mu_{2}}{q}\|v\|_{q}^{q}\,t^{\frac{(q-2)N}{2}}-\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}t^{\frac{(r_{1}+r_{2}-2)N}{2}}(1+s)^{r_{1}},$

we have

$\displaystyle\frac{d}{ds}J[t\star(1+s)u,t\star v]=$ $\displaystyle\|\nabla
u\|_{2}^{2}(1+s)t^{2}-\mu_{1}\|u\|_{p}^{p}(1+s)^{p-1}\,t^{\frac{(p-2)N}{2}}$
$\displaystyle-\beta
r_{1}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x(1+s)^{r_{1}-1}\,t^{\frac{(r_{1}+r_{2}-2)N}{2}}$
$\displaystyle+\left(\|\nabla u\|_{2}^{2}(1+s)^{2}+\|\nabla
v\|_{2}^{2}\right)t\,t^{\prime}-\frac{(p-2)N}{2p}\mu_{1}\|u\|_{p}^{p}(1+s)^{p}\,t^{\frac{(p-2)N-2}{2}}\,t^{\prime}$
$\displaystyle-\frac{(q-2)N}{2q}\mu_{2}\|v\|_{q}^{q}t^{\frac{(q-2)N-2}{2}}\,t^{\prime}$
$\displaystyle-\frac{(r_{1}+r_{2}-2)N}{2}\beta\Big{(}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\Big{)}(1+s)^{r_{1}}\,t^{\frac{(r_{1}+r_{2}-2)N-2}{2}}\,t^{\prime},$

here $t=t(s),t^{\prime}=t^{\prime}(s)$. Putting $s=0$ and noting that
$t(0)=1$, we have that

$\displaystyle\frac{d}{ds}J[t(s)\star(1+s)u,t(s)\star v]\Big{|}_{s=0}=$
$\displaystyle\big{[}\|\nabla u\|_{2}^{2}-\mu_{1}\|u\|_{p}^{p}-\beta
r_{1}\int_{\mathbb{R}^{N}}|u|^{r_{1}}|v|^{r_{2}}\mathrm{d}x\big{]}+P[u,v]t^{\prime}(0)$
$\displaystyle=$ $\displaystyle-\lambda_{1}\|u\|_{2}^{2}.$

Then

$C_{(a,b)}\leq J[t(s)\star(1+s)u,t(s)\star v]<J[u,v]=C_{(a,b)},\hbox{ for
$s>0$ small enough},$

a contradiction. Similarly, using that $\lambda_{2}>0$, we can prove that
$v\in S_{b}$. ∎

At this point we can give the

Proof of Theorem 2.2: Under the assumptions, by Lemma 7.3, we have that

$C_{(a,b)}<\min\\{m_{p,\mu_{1},a},m_{q,\mu_{2},b}\\}.$

Let $\\{[u_{n},v_{n}]\\}\subset\mathcal{H}$ be a Palais-Smale sequence as
obtained in Section 6. By Theorem 8.1, $[u_{n},v_{n}]\rightarrow[u,v]\in
S_{a}\times S_{b}$ and, in particular, $[u,v]\in\mathcal{P}_{a,b}$ and
$J[u,v]=C_{(a,b)}$. Invoking Remark 5.2 we deduce that $J$ restricted to
$\mathcal{P}_{(a,b)}$ admits a minimum, say $[\overline{u},\overline{v}]$
which consists in Schwarz symmetric functions. The fact that this minimum is a
critical point of $J$ constrained to $D_{a}\times D_{b}$ follows from Lemmas
4.8 and 4.9. As in the proof of Theorem 8.1 one deduces that the associated
Lagrange multipliers $\overline{\lambda}_{1}$ and $\overline{\lambda}_{2}$ are
both strictly positive. At this point we have proved that
$(\overline{\lambda}_{1},\overline{\lambda}_{2},\overline{u},\overline{v})\in\mathbb{R}^{2}\times\mathcal{H}$
is a ground state to Problem (1.4) having the required symmetry properties.
$\Box$

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