Abstract
The paper deals with the equilibrium solutions of twoequation systems in which each component equation has a variational structure. Solutions are obtained that can be in one of the following generalized Nash situations: (a) one component of the solution represents a mountain pass type critical point and the other is a minimizer; (b) both components of the solution are mountain pass type; (c) both components are minimizers, that is, the solution is a proper Nash equilibrium. The simultaneous treatment of critical points of the mountain pass type and of the minimum ones is achieved by using a unifying notion of linking. The theory is applied to a system of four elliptic equations in which the subsystems formed by the first two and the last two equations, respectively, are of gradient type. An example shows that the conditions found are noncontradictory. The theory could be applied to other classes of systems.
Authors
Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, BabeşBolyai University, ClujNapoca, Romania Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Andrei Stan
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Mathematics, BabeşBolyai University, ClujNapoca, Romania
Keywords
Variational method; linking; critical point; mountain pass geometry; Nash type equilibrium; monotone operator; elliptic system
Paper coordinates
R. Precup, A. Stan, Linking methods for componentwise variational systems, Results Math. 78 (2023) 246, https://doi.org/10.1007/s0002502302026x
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Paper (preprint) in HTML form
Linking methods for componentwise variational systems
Abstract
The paper deals with the equilibrium solutions of twoequation systems in which each component equation has a variational structure. Solutions are obtained that can be in one of the following generalized Nash situations: (a) one component of the solution represents a mountain pass type critical point and the other is a minimizer; (b) both components of the solution are mountain pass type; (c) both components are minimizers, that is, the solution is a proper Nash equilibrium. The simultaneous treatment of critical points of the mountain pass type and of the minimum ones is achieved by using a unifying notion of linking. The theory is applied to a system of four elliptic equations in which the subsystems formed by the first two and the last two equations, respectively, are of gradient type. An example shows that the conditions found are noncontradictory. The theory could be applied to other classes of systems.
keywords:
variational method, linking, critical point, mountain pass geometry, Nash type equilibrium, monotone operator, elliptic system.1 Introduction and Preliminaries
Numerous models that mathematically express realworld processes are represented as systems of equations. In certain circumstances, the solutions to these systems are determined to be critical points of a functional, which is dependent on the variables present within the system. In this scenario, is said that the system possesses a variational form. Thus, in such a case, assuming for simplicity only two variables ${u}_{1}$ and ${u}_{2},$ the system reads equivalently as
$$\{\begin{array}{c}{E}_{{u}_{1}}({u}_{1},{u}_{2})=0\hfill \\ {E}_{{u}_{2}}({u}_{1},{u}_{2})=0,\hfill \end{array}$$ 
where ${E}_{{u}_{1}},{E}_{{u}_{2}}$ are the partial derivatives of $E({u}_{1},{u}_{2})$ in each of the two variables. A wide range of variational techniques that are wellestablished in the literature are applicable to systems of this type. These techniques can be used to determine solutions as a minimizer or as a mountain pass type point of the functional $E({u}_{1},{u}_{2})$
In the present paper we are dealing with systems that do not have a variational form, but each of the component equations does. To be more precise, we examine two functionals, ${E}_{1}({u}_{1},{u}_{2})$ and ${E}_{2}({u}_{1},{u}_{2})$, and aim to find solutions to the system
$$\{\begin{array}{c}{E}_{11}({u}_{1},{u}_{2})=0\hfill \\ {E}_{22}({u}_{1},{u}_{2})=0,\hfill \end{array}$$  (1) 
where ${E}_{11}$ stands for the partial derivative of ${E}_{1}$ with respect to first variable and ${E}_{22}$ is the derivative of ${E}_{2}$ with respect to the second variable. It is natural to look for a solution $({u}_{1},{u}_{2})$ in one of the following situations:
 (a)

The pair $({u}_{1},{u}_{2})$ is a Nash (minmin) equilibrium of the system, that is ${u}_{1}$ minimizes the functional ${E}_{1}(\cdot ,{u}_{2})$ and ${u}_{2}$ minimizes ${E}_{2}({u}_{1},\cdot );$
 (b)

The pair $({u}_{1},{u}_{2})$ is a minmountain pass equilibrium of the system, that is ${u}_{1}$ minimizes the functional ${E}_{1}(\cdot ,{u}_{2})$ and ${u}_{2}$ is a mountain pass type point of ${E}_{2}({u}_{1},\cdot );$
 (c)

The pair $({u}_{1},{u}_{2})$ is a mountain passmountain pass equilibrium of the system, that is ${u}_{1}$ is a mountain pass type point of ${E}_{1}(\cdot ,{u}_{2})$ and ${u}_{2}$ is a mountain pass type point of ${E}_{2}({u}_{1},\cdot ).$
To have a simple understanding of these situations, it is enough to look at the functions on ${\mathbb{R}}^{2}\times {\mathbb{R}}^{2}$ by taking ${u}_{1}=(x,y)$ and ${u}_{2}=(z,w):$
(a)
${E}_{1}(x,y,z,w)$  $=$  ${x}^{2}+{y}^{2}+{z}^{2}+{w}^{2}xz,$  
${E}_{2}(x,y,z,w)$  $=$  ${x}^{2}+2{y}^{2}+{z}^{2}+{w}^{2}yw.$ 
It is easy to see that ${u}_{1}=(0,0)$ and ${u}_{2}=(0,0)$ solves (1) and that ${u}_{1}$ minimizes ${E}_{1}(\cdot ,{u}_{2})={x}^{2}+{y}^{2},$ while ${u}_{2}$ minimizes ${E}_{2}({u}_{1},\cdot )={z}^{2}+{w}^{2}.$
(b)
${E}_{1}(x,y,z,w)$  $=$  ${x}^{2}+{y}^{2}+{z}^{2}+{w}^{2}xz,$  
${E}_{2}(x,y,z,w)$  $=$  ${x}^{2}+2{y}^{2}+{z}^{2}{w}^{2}yw.$ 
Here again, ${u}_{1}=(0,0)$ and ${u}_{2}=(0,0)$ solves (1), and ${u}_{1}$ minimizes ${E}_{1}(\cdot ,{u}_{2})={x}^{2}+{y}^{2},$ while ${u}_{2}$ is a mountain pass of ${E}_{2}({u}_{1},\cdot )={z}^{2}{w}^{2}.$
(c)
${E}_{1}(x,y,z,w)$  $=$  ${x}^{2}{y}^{2}+{z}^{2}+{w}^{2}xz,$  
${E}_{2}(x,y,z,w)$  $=$  ${x}^{2}+2{y}^{2}+{z}^{2}{w}^{2}yw.$ 
In this case ${u}_{1}=(0,0)$ is a mountain pass type point of ${E}_{1}(\cdot ,{u}_{2})={x}^{2}{y}^{2}$ and ${u}_{2}=(0,0)$ is a mountain pass type point of ${E}_{2}({u}_{1},\cdot )={z}^{2}{w}^{2}.$
Our aim is to treat these three situations in an unitary way. This is possible thanks to the new notion of linking recently introduced in paper (20). It allows to produce both minimizers and mountain pass type critical points of a functional through the use of the same minmax method, where the distinction between the two is solely dependent on the type of linking employed.
The linking concept in critical point theory (see (3), (5), (23), (24), (27)) has its origin in the geometric condition of the mountain pass theorem due to Ambrosetti and Rabinowitz (1), and has undergone expansions along with the generalizations given to this theorem, becoming a successful tool in the study of many classes of nonlinear problems (see, e.g., (4), (6), (7), (9), (10), (11), (12), (15), (16), (22), (25)).
Our work uses the unifying notion of linking introduced in (20) and which we present in the following.
1.1 A unifying notion of linking
Let $X$ be a Banach space, $D$ and $Q$ be two subsets of $X$ with $\mathrm{\varnothing}\ne Q\subset D.$
Definition 1 ((20)).
We say that a nonempty set $A\subset D$ links a set $B\subset Q$ via $Q$ (in $D)$ if $\gamma \left(Q\right)\cap A\ne \mathrm{\varnothing}$ for every $\gamma \in C(Q,D)$ with ${\gamma }_{B}=$id${}_{B}.$
Note that, in virtue of the above definition, the total set $A=D$ links the empty set $B=\mathrm{\varnothing},$ via any $Q,$ in particular via any singleton $Q=\left\{\overline{u}\right\}$ with $\overline{u}\in D.$ As explained below, this limit case of (trivial) linking provides us with minima of a functional after applying the minmax procedure.
Assume that $A$ links $B$ in $D$ via $Q,$ let $\mathrm{\Gamma}=\{\gamma \in C(Q,D):\gamma {}_{B}=$id${}_{B}{}^{}\},$ and $E:D\to \mathbb{R}$ be any functional. Denote
$$m:=\underset{v\in D}{inf}E\left(v\right),a:=\underset{v\in A}{inf}E\left(v\right),b:=\underset{v\in B}{sup}E\left(v\right),$$ 
and
$$c:=\underset{\gamma \in \mathrm{\Gamma}}{inf}\underset{q\in Q}{sup}E\left(\gamma \left(q\right)\right).$$ 
Then it is easy to see that
$$m\le a\le c\phantom{\rule{1.5em}{0ex}}\text{and}b\le c.$$ 
If $B=\mathrm{\varnothing}$ and $A=D,$ then
$$m=a,b=\mathrm{\infty}\phantom{\rule{1em}{0ex}}\text{and}c=m.$$ 
The first equalities are obvious. To prove that $c=m,$ observe that in this case of trivial linking, $\mathrm{\Gamma}$ is the whole space $C(Q,D)$ and for any constant mapping ${\gamma}_{v}\left(q\right)=v$ for all $q\in Q,$ with $v\in D,$ one has ${sup}_{q\in Q}E\left(\gamma \left(q\right)\right)=E\left(v\right)$ and then
$$c=\underset{\gamma \in \mathrm{\Gamma}}{inf}\underset{q\in Q}{sup}E\left(\gamma \left(q\right)\right)\le \underset{v\in D}{inf}\underset{q\in Q}{sup}E\left({\gamma}_{v}\left(q\right)\right)=\underset{v\in D}{inf}E\left(v\right)=m.$$ 
The converse inequality being obvious, it follows that $c=m$ as claimed. Therefore, the adopted definition of linking allows us to treat the minimization of a functional $E$ on a set $D$ as a minimax problem and thus to make no distinction between the minimax problems and the minimization ones.
In this paper, we consider two functionals of two variables, ${E}_{1}({u}_{1},{u}_{2})$ and ${E}_{2}({u}_{1},{u}_{2}),$ defined on a product space ${X}_{1}\times {X}_{2}.$ Correspondingly, we shall use one linking for the functionals ${E}_{1}(\cdot ,{u}_{2})$ with a fixed ${u}_{2}\in {X}_{2},$ and an other linking for the functionals ${E}_{2}({u}_{1},\cdot )$ when ${u}_{1}$ is fixed in ${X}_{1}.$ Depending on the type of the two linkings, trivial or nontrivial, we shall reach one of the above situations (a), (b), or (c).
We conclude this introductory section by some additional tools which are used.
1.2 Ekeland variational principle
The proof of our main result is essentially based on the weak form of Ekeland’s variational principle (see, e.g., (8)).
Lemma 1 (Ekeland Principle  weak form).
Let $(X,d)$ be a complete metric space and let $\mathrm{\Phi}:X\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ be a lower semicontinuous and bounded below functional. Then, given any $\epsilon >0$, there exists ${u}_{\epsilon}\in X$ such that
$$\mathrm{\Phi}({u}_{\epsilon})\le \underset{X}{inf}\mathrm{\Phi}+\epsilon $$ 
and
$$\mathrm{\Phi}({u}_{\epsilon})\le \mathrm{\Phi}(u)+\epsilon d(u,{u}_{\epsilon}),$$ 
for all $u\in X.$
1.3 Two auxiliary results
The first lemma is used together with Ekeland’s principle in the proof of our first main result in Section 2.
Lemma 2.
Let $(X,\cdot {}_{X})$ be a Banach space, $K$ a compact and $f\in C(K,{X}^{\ast}).$ Then, for each $\epsilon >0$, there exists a function $\phi \in C(K,X)$ such that:
$${\left\phi \left(x\right)\right}_{X}\le 1,\text{and}(f\left(x\right),\phi \left(x\right)){\leftf\left(x\right)\right}_{X}\epsilon ,$$ 
for all $x\in K$.
Proof.
A direct consequence of the definition of the dual norm is that, for any ${x}^{\ast}\in {X}^{\ast}$ and $\epsilon >0$, there exists an ${x}_{\epsilon}\in X$ that satisfies
$${{x}_{\epsilon}}_{X}\le 1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}({x}^{*},{x}_{\epsilon})>{x}_{\epsilon}\epsilon .$$ 
Let $\epsilon >0$. According to the previous remark, for any ${x}_{0}\in K$, there is ${u}_{0}\in X$ with ${\left{u}_{0}\right}_{X}\le 1$ and $(f\left({x}_{0}\right),{u}_{0})>\leftf\left({x}_{0}\right)\right\epsilon .$ Define
$$U\left({x}_{0}\right)=\{x\in K:(f\left(x\right),{u}_{0})>{\leftf\left(x\right)\right}_{X}\epsilon \},$$ 
and note that $U\left({x}_{0}\right)$ is open in $K$ and ${x}_{0}\in U\left({x}_{0}\right)$. Since $K={\cup}_{x\in K}U\left(x\right),$ there is a finite open covering of $K:$ $U\left({x}_{1}\right),U\left({x}_{2}\right),\mathrm{\dots},U\left({x}_{n}\right).$ Let ${u}_{i}$ $(i=1,2,\mathrm{\dots},n)$ be the corresponding elements, i.e.,
$$U\left({x}_{i}\right)=\{x\in K:(f\left(x\right),{u}_{i})>{\leftf\left(x\right)\right}_{X}\epsilon \}.$$ 
Let ${\rho}_{i}\left(x\right)=$dist $(x,K\setminus U\left({x}_{i}\right))$ and ${\zeta}_{i}\left(x\right)={\rho}_{i}\left(x\right)/{\sum}_{j=1}^{n}{\rho}_{j}\left(x\right).$ Notice that ${\zeta}_{i}:K\to K$ is continuous, ${\zeta}_{i}\left(x\right)\ne 0$ if and only if $x\in U\left({x}_{i}\right)$ and ${\sum}_{j=1}^{n}$ ${\zeta}_{i}\left(x\right)=1$, for all $x\in K.$ Finally, the desired function is
$$\phi \left(x\right)=\sum _{j=1}^{n}{\zeta}_{i}\left(x\right){x}_{i}.$$ 
∎
The second lemma concerns the convergence to zero of two sequences of nonnegative numbers that satisfy a comparison inequality in matrix form.
Lemma 3.
Let ${\left({x}_{k,p}\right)}_{k\ge 1},{\left({y}_{k,p}\right)}_{k\ge 1}$ be two sequences of nonnegative real numbers depending on a parameter $p$ such that
$$\left[\begin{array}{c}{x}_{k,p}\\ {y}_{k,p}\end{array}\right]\le A\left[\begin{array}{c}0\\ {y}_{k1,p}\end{array}\right]+\left[\begin{array}{c}{z}_{k,p}\\ {w}_{k,p}\end{array}\right],$$ 
for all $k$ and $p,$ where ${\left({z}_{k,p}\right)}_{k\ge 1},{\left({w}_{k,p}\right)}_{k\ge 1}$ are sequences convergent to zero uniformly with respect to $p$. If the matrix $A$ is convergent to zero and the sequence ${\left({y}_{k,p}\right)}_{k\ge 1}$ is bounded uniformly with respect to $p$, then ${x}_{k,p}\to 0$ and ${y}_{k,p}\to 0$ as $k\to \mathrm{\infty}$ uniformly with respect to $p.$
The proof is similar to the one in (26, Lemma 2.2).
Recall that a square matrix $A\in {\mathcal{M}}_{n\times n}\left({\mathbb{R}}_{+}\right)$ is said to be convergent to zero if its power ${A}^{k}$ tends to the zero matrix as $k\to \mathrm{\infty}.$ The same conclusion holds if the spectral radius of the matrix is less than one, or if the inverse of $IA$ (where $I$ is the identity matrix) is both invertible and has nonnegative entries.
In particular, for $n=2,$ a matrix $A\in {\mathcal{M}}_{2\times 2}\left({\mathbb{R}}_{+}\right)$ is convergent to zero if and only if
$$ 
Also, one can easily check that if $A={\left[{a}_{ij}\right]}_{1\le i,j\le 2}$ is convergent to zero and
$${A}^{\prime}:=\left[\begin{array}{cc}{a}_{11}& 0\\ {a}_{21}& {a}_{22}\end{array}\right],{A}^{\prime \prime}:=A{A}^{\prime}=\left[\begin{array}{cc}0& {a}_{12}\\ 0& 0\end{array}\right],$$  (2) 
then $I{A}^{\prime}$ is invertible and the matrix ${\left(I{A}^{\prime}\right)}^{1}{A}^{\prime \prime}$ is also convergent to zero.
The paper is structured as follows: Section 2 contains the abstract results about the existence solution of system (1) in a Hilbert space, which fall under one of the three scenarios (a), (b), or (c) depending on the linking type. The proofs make use of Ekeland’s principle, and monotonicity type properties related to the derivatives of the two functionals. Section 3 is devoted to an application to a coupled system of four elliptic equations subject to the homogeneous Dirichlet condition.
The paper substantially complements the paper (20) and expands the ideas and working techniques from (2) and our previous works (17; 18; 19; 21; 26) (see also (13, Ch. 8)). But the absolute novelty brought by this work consists in obtaining solutions of some nonlinear systems which, relative to the associated energy functionals, are generalized Nashtype equilibria, in the sense that some of the components of the solution can be mountain pass critical points, and others minimum points. The theory, although presented in the case of systems with two equations, can be extended to systems with any number of equations.
2 Main results
Let ${H}_{i}$ $\left(i=1,2\right)$ be Hilbert spaces with inner product ${(\cdot ,\cdot )}_{i}$ and norm $\cdot {}_{i}$ which are identified with their duals, and denote $H={H}_{1}\times {H}_{2}.$ For each space ${H}_{i}$, consider a linking giving by two closed sets ${A}_{i},{B}_{i}\subset {H}_{i}$ and a compact set ${Q}_{i}\subset {H}_{i}$ with ${A}_{i},{Q}_{i}\ne \mathrm{\varnothing}$ and ${B}_{i}\subset {Q}_{i}.$ Denote
$${\mathrm{\Gamma}}_{i}:=\{{\gamma}_{i}\in C({Q}_{i},{H}_{i}):{\gamma}_{i}({u}_{i})={u}_{i}\text{for all}{u}_{i}\in {B}_{i}\}.$$ 
One sees that these sets are complete metric spaces together with the metric ${d}_{i},$ given by
$${d}_{i}({\gamma}_{i},\overline{{\gamma}_{i}}):=\underset{q\in {Q}_{i}}{\mathrm{max}}{\gamma}_{i}(q)\overline{{\gamma}_{i}}(q)ver{t}_{i},$$ 
for any ${\gamma}_{i},\overline{{\gamma}_{i}}\in {\mathrm{\Gamma}}_{i}.$ Furthermore, for two functionals ${E}_{i}:H\to \mathbb{R}$ and each point $({u}_{1},{u}_{2})\in H,$ we define:
$$\begin{array}{cc}{m}_{1}({u}_{2}):={inf}_{{X}_{1}}{E}_{1}(\cdot ,{u}_{2});\hfill & {m}_{2}({u}_{1}):={inf}_{{X}_{2}}{E}_{2}({u}_{1},\cdot );\hfill \\ {a}_{1}({u}_{2}):={inf}_{{A}_{1}}{E}_{1}(\cdot ,{u}_{2});\hfill & {a}_{2}({u}_{1}):={inf}_{{A}_{2}}{E}_{2}({u}_{1},\cdot );\hfill \\ {b}_{1}({u}_{2}):={sup}_{{B}_{1}}{E}_{1}(\cdot ,{u}_{2});\hfill & {b}_{2}({u}_{1}):={sup}_{{B}_{2}}{E}_{2}({u}_{1},\cdot );\hfill \end{array}$$ 
${c}_{1}({u}_{2}):=\underset{\mu \in {\mathrm{\Gamma}}_{1}}{inf}\underset{q\in {Q}_{1}}{\mathrm{max}}{E}_{1}(\mu (q),{u}_{2});$  
${c}_{2}({u}_{1}):=\underset{\mu \in {\mathrm{\Gamma}}_{2}}{inf}\underset{q\in {Q}_{2}}{\mathrm{max}}{E}_{2}({u}_{1},\mu (q)).$ 
As noted above, for each $i\in \{1,2\},$ one has
$${m}_{i}\le {a}_{i}\le {c}_{i}\phantom{\rule{1em}{0ex}}\text{and}{b}_{i}\le {c}_{i}.$$ 
Assume that ${E}_{i}$ $\left(i=1,2\right)$ is a ${C}^{1}$ functional on $H$ and denote by ${E}_{ii}$ the partial Fréchet derivative of ${E}_{i}$ with respect to the $i$th variable.
Our first result is the following theorem.
Theorem 4.
For each $i\in \{1,2\},$ let ${A}_{i}$ links ${B}_{i}$ via ${Q}_{i}$ in ${H}_{i}.$ If
$$ 
then there exist two sequences ${u}_{1}^{k}\in {H}_{1}$ and ${u}_{2}^{k}\in {H}_{2}$ such that
$$0\le {E}_{1}({u}_{1}^{k},{u}_{2}^{k1}){c}_{1}\left({u}_{2}^{k1}\right)\to 0,0\le {E}_{2}({u}_{1}^{k},{u}_{2}^{k}){c}_{2}\left({u}_{1}^{k}\right)\to 0$$  (3) 
and
$${E}_{11}({u}_{1}^{k},{u}_{2}^{k1})\to 0,{E}_{22}({u}_{1}^{k},{u}_{2}^{k})\to 0,$$  (4) 
as $k\to \mathrm{\infty}.$
Proof.
We shall construct the two desired sequences $\left({u}_{1}^{k}\right),\left({u}_{2}^{k}\right)$ by an iterative procedure working alternatively on the two functionals. We start with an arbitrary point ${v}_{0}\in {H}_{2}$. We follow two stages:
(a) first consider the functional ${\mathcal{E}}_{1}:{\mathrm{\Gamma}}_{1}\to \mathbb{R},$
$${\mathcal{E}}_{1}(\mu )=\underset{{Q}_{1}}{\mathrm{max}}{E}_{1}(\mu (\cdot ),{u}_{2}^{k1})\phantom{\rule{1em}{0ex}}\left(\mu \in {\mathrm{\Gamma}}_{1}\right),$$ 
and observe that it is semicontinuous and bounded from below, since
$${\mathcal{E}}_{1}({\gamma}_{1})\ge {a}_{1}({u}_{2}^{k1})>{b}_{1}\left({u}_{2}^{k1}\right)\ge \mathrm{\infty}$$ 
Thus, Lemma 1 guarantees the existence of a path ${\gamma}_{1}^{k}\in {\mathrm{\Gamma}}_{1}$ such that
$${\mathcal{E}}_{1}({\gamma}_{1}^{k})\le \underset{\mu \in {\mathrm{\Gamma}}_{1}}{inf}{\mathcal{E}}_{1}(\mu )+\frac{1}{k}={c}_{1}\left({u}_{2}^{k1}\right)+\frac{1}{k},$$  (5) 
$${\mathcal{E}}_{1}({\gamma}_{1}^{k}){\mathcal{E}}_{1}(\mu )\le \frac{1}{k}{d}_{1}({\gamma}_{1}^{k},\mu ),$$  (6) 
for all $\mu \in {\mathrm{\Gamma}}_{1}.$ If we consider
$${Q}_{1}^{k}:=\{{q}_{1}\in {Q}_{1}:{\mathcal{E}}_{1}\left({\gamma}_{1}^{k}({q}_{1})\right)={E}_{1}({\gamma}_{1}^{k}({q}_{1}),{u}_{2}^{k1})\},$$ 
one can see that ${B}_{1}\cap {Q}_{1}^{k}=\mathrm{\varnothing}$, since $$
Next we prove that there exists ${q}_{1}^{k}\in {Q}_{1}^{k}$ with $$ To this end we apply Lemma 2 to the function
$$f\left({q}_{1}\right)={E}_{11}({\gamma}_{1}^{k}\left(\mu \right),{u}_{2}^{k1}),$$ 
from where we deduce the existence of a function $\phi \in C({Q}_{1},{H}_{1})$ with ${\left\phi \left({q}_{1}\right)\right}_{1}\le 1$ and
$${({E}_{11}({\gamma}_{1}^{k}\left({q}_{1}\right),{u}_{2}^{k1}),\phi \left({q}_{1}\right))}_{1}>{\left{E}_{11}({\gamma}_{1}^{k}\left({q}_{1}\right),{u}_{2}^{k1})\right}_{1}\frac{1}{k}\phantom{\rule{1.5em}{0ex}}\text{on}{Q}_{1}.$$  (7) 
In (5) take $\eta ={\gamma}_{1}^{k}\lambda w$ with $\lambda >0$ and
$$w\left({q}_{1}\right)=\zeta \left({q}_{1}\right)\phi \left({q}_{1}\right),$$ 
where $\zeta :{Q}_{1}\to [0,1]$ is continuous, $\zeta \left({q}_{1}\right)=1$ on ${Q}_{1}^{k}$ and $\zeta =0$ on ${B}_{1}.$ We have ${d}_{1}({\gamma}_{1}^{k},\eta )=\lambda {\leftw\right}_{\mathrm{\infty}}\le \lambda $ and
$$\psi \left(\eta \right)=\underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}(\eta \left({q}_{1}\right),{u}_{2}^{k1})={E}_{1}(\eta \left({q}_{1}^{\lambda}\right),{u}_{2}^{k1}),$$ 
for some ${q}_{1}^{\lambda}\in {Q}_{1}.$ Hence from (6), one has
$${E}_{1}(\eta \left({q}_{1}^{\lambda}\right),{u}_{2}^{k1})\underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}({\gamma}_{1}^{k}\left({q}_{1}\right),{u}_{2}^{k1})+\frac{\lambda}{k}\ge 0.$$ 
Since
${E}_{1}(\eta \left({q}_{1}^{\lambda}\right),{u}_{2}^{k1}){E}_{1}({\gamma}_{1}^{k}\left({q}_{1}^{\lambda}\right),{u}_{2}^{k1})$  
$=\lambda {({E}_{11}({\gamma}_{1}^{k}\left({q}_{1}^{\lambda}\right),{u}_{2}^{k1}),w\left({q}_{1}^{\lambda}\right))}_{1}+o\left(\lambda \right)$ 
we deduce that
$${({E}_{11}({\gamma}_{1}^{k}\left({q}_{1}^{\lambda}\right),{u}_{2}^{k1}),w\left({q}_{1}^{\lambda}\right))}_{1}+\frac{1}{k}+\frac{1}{\lambda}o\left(\lambda \right)\ge 0.$$ 
We may assume that ${q}_{1}^{\lambda}\to {q}_{1}^{k}\in {Q}_{1}^{k}$ as $\lambda \to 0.$ Then
$${({E}_{11}({\gamma}_{1}^{k}\left({q}_{1}^{k}\right),{u}_{2}^{k1}),w\left({q}_{1}^{k}\right))}_{1}+\frac{1}{k}\ge 0.$$ 
Thus, also using (7) and since $w\left({q}_{1}^{k}\right)=\phi \left({q}_{1}^{k}\right),$ we have
$$ 
whence
$$ 
We denote
$${u}_{1}^{k}={\gamma}_{1}^{k}\left({q}_{1}^{k}\right).$$ 
Thus we have
$$  (8) 
(b) Now using the element ${u}_{1}^{k},$ we proceed to construct ${u}_{2}^{k}.$ To this aim we follow a similar strategy for the functional ${\mathcal{E}}_{2}:{\mathrm{\Gamma}}_{2}\to \mathbb{R},$
$${\mathcal{E}}_{2}(\mu )=\underset{{q}_{2}\in {Q}_{2}}{\mathrm{max}}{E}_{2}({u}_{1}^{k},\mu \left({q}_{2}\right))\phantom{\rule{1em}{0ex}}\left(\mu \in {\mathrm{\Gamma}}_{2}\right).$$ 
In the end we obtain an element ${u}_{2}^{k}\in {H}_{2}$ of the form
$${u}_{2}^{k}={\gamma}_{2}^{k}\left({q}_{2}^{k}\right)$$ 
with ${\gamma}_{2}^{k}\in {\mathrm{\Gamma}}_{2}$ and
$${q}_{2}^{k}\in {Q}_{2}^{k}=\{{q}_{2}\in {Q}_{2}:{\mathcal{E}}_{2}\left({\gamma}_{2}^{k}({q}_{2})\right)={E}_{2}({u}_{1}^{k},{\gamma}_{2}^{k}({q}_{2}))\},$$ 
having the properties
$$  (9) 
∎
In the subsequent, we establish further proprieties of the sequences $\left({u}_{1}^{k}\right),\left({u}_{2}^{k}\right)$ constructed in the proof of Theorem 4.
Theorem 5.
If the sequences $\left({u}_{1}^{k}\right),\left({u}_{2}^{k}\right)$ are convergent, i.e., there exists ${u}^{\ast},{v}^{\ast}$ such that ${u}_{1}^{k}\to {u}^{\ast}$ and ${u}_{2}^{k}\to {v}^{\ast}$, then
$${E}_{11}({u}^{\ast},{v}^{\ast})=0,{E}_{22}({u}^{\ast},{v}^{\ast})=0,$$  (10) 
$${c}_{1}\left({u}_{2}^{k}\right)\to {c}_{1}\left({v}^{\ast}\right),{c}_{2}\left({u}_{1}^{k}\right)\to {c}_{2}\left({u}^{\ast}\right)$$  (11) 
and
$${E}_{1}({u}^{\ast},{v}^{\ast})={c}_{1}\left({v}^{\ast}\right),{E}_{2}({u}^{\ast},{v}^{\ast})={c}_{2}\left({u}^{\ast}\right).$$  (12) 
Proof.
Clearly, relation (10) follows directly from (4). Also, if (11) holds true, then we can easily derive relation (12) from (3). Thus, it remains us to prove (11).
We provide the conclusion for ${c}_{1}\left({u}_{2}^{k}\right)$, and the same can be deduced for ${c}_{2}\left({u}_{1}^{k}\right)$ through a similar process.
Step 1: ${c}_{1}\left({u}_{2}^{k1}\right)\to {E}_{1}({u}^{\ast},{v}^{\ast}).$ Indeed, one has
${c}_{1}\left({u}_{2}^{k1}\right)$  $=$  $\underset{\mu \in {\mathrm{\Gamma}}_{1}}{inf}\underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}(\mu \left({q}_{1}\right),{u}_{2}^{k1})\le \underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}({\gamma}_{1}^{k}\left({q}_{1}\right),{u}_{2}^{k1})$  
$=$  ${E}_{1}({\gamma}_{1}^{k}\left({q}_{1}^{k}\right),{u}_{2}^{k1})={E}_{1}({u}_{1}^{k},{u}_{2}^{k1})\le {c}_{1}\left({u}_{2}^{k1}\right)+{\displaystyle \frac{1}{k}}.$ 
Hence
${E}_{1}({u}_{1}^{k},{u}_{2}^{k1}){E}_{1}({u}^{\ast},{v}^{\ast}){\displaystyle \frac{1}{k}}$  $\le $  ${c}_{1}\left({u}_{2}^{k1}\right){E}_{1}({u}^{\ast},{v}^{\ast})$  
$\le $  ${E}_{1}({u}_{1}^{k},{u}_{2}^{k1}){E}_{1}({u}^{\ast},{v}^{\ast}),$ 
whence passing to the limit we deduce that ${c}_{1}\left({u}_{2}^{k1}\right){E}_{1}({u}^{\ast},{v}^{\ast})\to 0,$ as claimed.
Step 2: ${E}_{1}({u}^{\ast},{v}^{\ast})\le {c}_{1}\left({v}^{\ast}\right).$ Let $\mu \in {\mathrm{\Gamma}}_{1}$ be any path. Then for each $k,$ there is ${\overline{q}}_{1}^{k}\in {Q}_{1}$ with
$${c}_{1}\left({u}_{2}^{k1}\right)\le \underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}(\mu \left({q}_{1}\right),{u}_{2}^{k1})={E}_{1}(\mu \left({\overline{q}}_{1}^{k}\right),{u}_{2}^{k1}).$$ 
Since ${Q}_{1}$ is compact, passing to a subsequence we may assume that ${\overline{q}}_{1}^{k}\to {q}_{1}^{\mu}$ as $k\to \mathrm{\infty}.$ Then taking the limit in the last inequality and using the conclusion from Step 1, we derive
$${E}_{1}({u}^{\ast},{v}^{\ast})\le {E}_{1}(\mu \left({\overline{q}}_{1}^{\mu}\right),{v}^{\ast})\le \underset{{q}_{1}\in {Q}_{1}}{\mathrm{max}}{E}_{1}(\mu \left({q}_{1}\right),{v}^{\ast}),$$ 
whence taking the infimum over $\mu \in {\mathrm{\Gamma}}_{1}$ we obtain the desired inequality.
Step 3: ${E}_{1}({u}^{\ast},{v}^{\ast})\ge {c}_{1}\left({v}^{\ast}\right).$ From the definition of ${c}_{1},$ one clearly has ${c}_{1}\left({v}^{\ast}\right)\le {E}_{1}({\gamma}_{1}^{k}\left({q}_{1}^{k}\right),{v}^{\ast})={E}_{1}({u}_{1}^{k},{v}^{\ast})$ for all $k.$ Let $\epsilon >0$ be arbitrarily chosen. Since ${u}_{2}^{k}\to {v}^{\ast},$ there exists ${j}_{k}$ such that ${c}_{1}\left({v}^{\ast}\right)\epsilon \le {E}_{1}({u}_{1}^{k},{v}_{j})$ for all $j\ge {j}_{k}.$ Thus, we can assume that ${j}_{k}>{j}_{k1}$ and so that ${j}_{k}\to \mathrm{\infty}$ as $k\to \mathrm{\infty}.$ Then, from
$${c}_{1}\left({v}^{\ast}\right)\epsilon \le {E}_{1}({u}_{1}^{k},{v}_{{j}_{k}}),$$ 
letting $k$ go to infinity, we deduce
$${c}_{1}\left({v}^{\ast}\right)\epsilon \le {E}_{1}({u}^{\ast},{v}^{\ast}).$$ 
Now since $\epsilon $ is arbitrary, we must have ${c}_{1}\left({v}^{\ast}\right)\le {E}_{1}({u}^{\ast},{v}^{\ast}),$ as claimed.
Finally, the two contrary inequalities in Steps 2 and 3 show that ${c}_{1}\left({v}^{\ast}\right)={E}_{1}({u}^{\ast},{v}^{\ast}).$ ∎
Remark 1.
In the light of the conclusions of Theorem 5, we can distinguish the following situations:
(a) If both linkings of the spaces ${H}_{1}$ and ${H}_{2}$ are trivial, then ${u}^{\ast}$ is a minimizer of the functional ${E}_{2}(\cdot ,{v}^{\ast})$ and ${v}^{\ast}$ is a minimizer of the functional ${E}_{2}({u}^{\ast},\cdot ),$ that is the couple $({u}^{\ast},{v}^{\ast})$ is a Nash equilibrium of the functionals ${E}_{1},{E}_{2}.$
(b) If only the linking of the space ${H}_{2}$ is the trivial one, then ${u}^{\ast}$ is a mountain pass type point of ${E}_{1}(\cdot ,{v}^{\ast}),$ while ${v}^{\ast}$ is a minimizer of the functional ${E}_{2}({u}^{\ast},\cdot ).$
(b) If both linkings of the spaces ${H}_{1}$ and ${H}_{2}$ are nontrivial, then ${u}^{\ast}$ is a mountain pass type point of the functional ${E}_{2}(\cdot ,{v}^{\ast})$ and ${v}^{\ast}$ is a mountain pass type point of the functional ${E}_{2}({u}^{\ast},\cdot ).$
The next result answers the problem of convergence of sequences ${u}_{1}^{k}$ and ${u}_{2}^{k}.$ It requires some monotonicity conditions related to the derivatives ${E}_{11}$ and ${E}_{22}.$
Theorem 6.
Let $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ be the two sequences constructed in the proof of Theorem 4. Let $L=({L}_{1},{L}_{2}):H\to H,{L}_{i}:H\to {H}_{i}\left(i=1,2\right)$ be a continuous operator and let $N=({N}_{1},{N}_{2}):H\to H,$ ${N}_{i}:H\to {H}_{i}$ $\left(i=1,2\right),$ be defined by
$$N\left(u\right)=uL({E}_{11}\left(u\right),{E}_{22}\left(u\right)).$$  (13) 
Assume that the following conditions are satisfied:
(i) there are nonnegative constants ${a}_{ij}$ $\left(i,j=1,2\right)$ such that
${({N}_{1}({u}_{1},{u}_{2}){N}_{1}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{1}{\overline{u}}_{1})}_{1}$  (14)  
$\le {a}_{11}{{u}_{1}{\overline{u}}_{1}}_{1}^{2}+{a}_{12}{\left{u}_{1}{\overline{u}}_{1}\right}_{1}{\left{u}_{2}{\overline{u}}_{2}\right}_{2},$  
${({N}_{2}({u}_{1},{u}_{2}){N}_{2}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{2}{\overline{u}}_{2})}_{2}$  (15)  
$\le {a}_{22}{{u}_{2}{\overline{u}}_{2}}_{2}^{2}+{a}_{21}{\left{u}_{1}{\overline{u}}_{1}\right}_{1}{\left{u}_{2}{\overline{u}}_{2}\right}_{2},$ 
for all ${u}_{1},{\overline{u}}_{1}\in {H}_{1}$ and ${u}_{2},{\overline{u}}_{2}\in {H}_{2};$
(ii) the matrix $A={[{a}_{i,j}]}_{1\le i,j\le 2}$ is convergent to zero;
(iii) the sequence $\left({u}_{2}^{k}\right)$ (equivalently $\left({u}_{1}^{k}\right)$) is bounded.
Then the sequences $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ are convergent.
Proof.
Since the sequences ${E}_{11}({u}_{1}^{k},{u}_{2}^{k1}),{E}_{22}({u}_{1}^{k},{u}_{2}^{k})$ are convergent to zero, and the operators ${L}_{1},{L}_{2}$ are continuous, one has that the sequences
${\alpha}_{k}:={L}_{1}({E}_{11}({u}_{1}^{k},{u}_{2}^{k1}),{E}_{22}({u}_{1}^{k},{u}_{2}^{k}),$  
${\beta}_{k}:={L}_{1}({E}_{11}({u}_{1}^{k},{u}_{2}^{k1}),{E}_{22}({u}_{1}^{k},{u}_{2}^{k})$ 
are also convergent to zero. In terms of ${\alpha}_{k}$ and ${\beta}_{k},$ formula (13) gives
$${u}_{1}^{k}={\alpha}_{k}+{N}_{1}({u}_{1}^{k},{u}_{2}^{k1}),{u}_{2}^{k}={\beta}_{k}+{N}_{2}({u}_{1}^{k},{u}_{2}^{k}).$$ 
Then, using the monotony conditions (14), we deduce
${\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1}^{2}$  $={({u}_{1}^{k}{u}_{1}^{k+p},{\alpha}_{k}{\alpha}_{k+p})}_{1}$  (16)  
$+{({u}_{1}^{k}{u}_{1}^{k+p},{N}_{1}({u}_{1}^{k},{u}_{2}^{k1}){N}_{1}({u}_{1}^{k+p},{u}_{2}^{k+p1}))}_{1}$  
$\le {a}_{11}{\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1}^{2}+{a}_{12}{\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1}{\left{u}_{2}^{k1}{u}_{2}^{k+p1}\right}_{2}$  
$+{\left{\alpha}_{k}{\alpha}_{k+p}\right}_{1}{\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1}.$ 
Similarly,
${\left{u}_{2}^{k}{u}_{2}^{k+p}\right}_{2}^{2}$  $\le {a}_{22}{\left{u}_{2}^{k}{u}_{2}^{k+p}\right}_{2}^{2}+{a}_{21}{\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1}{\left{u}_{2}^{k}{u}_{2}^{k+p}\right}_{2}$  (17)  
$+{\left{\beta}_{k}{\beta}_{k+p}\right}_{2}{\left{u}_{2}^{k}{u}_{2}^{k+p}\right}_{2}.$ 
Under the notations
${x}_{k,p}:={\left{u}_{1}^{k}{u}_{1}^{k+p}\right}_{1},{y}_{k,p}={\left{u}_{2}^{k}{u}_{2}^{k+p}\right}_{2},$  
${z}_{k,p}:={\left{\alpha}_{k}{\alpha}_{k+p}\right}_{1},{w}_{k,p}:={\left{\beta}_{k}{\beta}_{k+p}\right}_{2},$ 
inequalities (16)(17) can be put under the matrix form
$$\left[\begin{array}{c}{x}_{k,p}\\ {y}_{k,p}\end{array}\right]\le {A}^{\prime}\left[\begin{array}{c}{x}_{k,p}\\ {y}_{k,p}\end{array}\right]+{A}^{\prime \prime}\left[\begin{array}{c}0\\ {y}_{k1,p}\end{array}\right]+\left[\begin{array}{c}{z}_{k,p}\\ {w}_{k,p}\end{array}\right],$$  (18) 
where the matrices ${A}^{\prime}$ and ${A}^{\prime \prime}$ are those from (2). One can see that (18) is equivalent to
$$\left[\begin{array}{c}{x}_{k,p}\\ {y}_{k,p}\end{array}\right]\le \stackrel{~}{A}\left[\begin{array}{c}0\\ {y}_{k1,p}\end{array}\right]+{\left(I{A}^{\prime}\right)}^{1}\left[\begin{array}{c}{z}_{k,p}\\ {w}_{k,p}\end{array}\right],$$ 
where the matrix $\stackrel{~}{A}:={(I{A}^{\prime})}^{1}{A}^{\prime \prime}$ is convergent to zero. Thus, Lemma 3 provides assurance that the sequences ${\left({x}_{k,p}\right)}_{k\ge 1}$ and ${\left({y}_{k,p}\right)}_{k\ge 1}$ converge to zero uniformly with respect to $p,$ i.e., $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ are Cauchy sequences, hence convergent. ∎
Remark 2.
To proceed with the limit transition in equations (3) and (4) it is crucial to ensure the convergence of the entire sequences $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ and not only some of their subsequences. This is due to the phase shift of the sequence $\left({u}_{2}^{k}\right)$ by one unit. Indeed, if a subsequence ${\left({u}_{2}^{{k}_{j}}\right)}_{j\ge 1}$ is convergent, then it is not sure that the sequence ${\left({u}_{2}^{{k}_{j}1}\right)}_{j\ge 1}$ also converges and has the same limit.
Remark 3.
By using a continuous operator $L,$ a continuous transformation of the derivatives is actually achieved, on which monotonicity conditions are imposed. Without such a transformation, monotony conditions seem to be impossible to fulfill given the nature of the mountain pass geometry. We mention that in our previous works devoted to Nashtype equilibria, it was possible to avoid using a special operator $L,$ which there, was actually the identity operator.
It remains to give sufficient conditions to guarantee the boundedness of the sequence $\left({u}_{2}^{k}\right).$
Theorem 7.
The sequence $\left({u}_{2}^{k}\right)$ is bounded in each one of the following situations:
(a) The linking in ${H}_{2}$ is trivial; for some $w\in {H}_{2},$
$${E}_{2}(\cdot ,w)\phantom{\rule{1em}{0ex}}\text{is bounded on}{H}_{1};\text{}$$  (19) 
$${E}_{2}(u,\cdot )\text{is coercive uniformly with respect to}u.$$  (20) 
(b) The linking in ${H}_{2}$ is nontrivial; for some $w\in {B}_{2},$
$${E}_{2}(\cdot ,w)\text{is bounded on}{H}_{1};$$  (21) 
$${E}_{2}(u,\cdot )\text{is coercive uniformly with respect to}u.$$  (22) 
Proof.
(a) The linking in ${H}_{2}$ being trivial, one has ${c}_{2}\left({u}_{1}^{k}\right)={m}_{2}\left({u}_{1}^{k}\right)$ and then from (9) and (19),
$${E}_{2}({u}_{1}^{k},{u}_{2}^{k})\le {m}_{2}\left({u}_{1}^{k}\right)+\frac{1}{k}\le {E}_{2}({u}_{1}^{k},w)+1\le C$$ 
for all $k$ and some constant $C.$ This, in virtue of (20), gives the conclusion.
(b) From (21), there is a constant $C$ with $C\le {E}_{2}({u}_{1}^{k},w)$ for all $k.$ Since $w\in {B}_{2},$ one has ${\gamma}_{2}^{k}\left(w\right)=w.$ Then
$C$  $\le $  ${E}_{2}({u}_{1}^{k},w)={E}_{2}({u}_{1}^{k},{\gamma}_{2}^{k}\left(w\right))\le \underset{{q}_{2}\in {Q}_{2}}{\mathrm{max}}{E}_{2}({u}_{1}^{k},{\gamma}_{2}^{k}\left({q}_{2}\right))$  
$=$  ${E}_{2}({u}_{1}^{k},{\gamma}_{2}^{k}\left({q}_{2}^{k}\right))={E}_{2}({u}_{1}^{k},{u}_{2}^{k}),$ 
which, in virtue of (22), gives the conclusion. ∎
We note that in applications, some other more specific conditions can be invoked in order to guarantee the boundedness of $\left({u}_{2}^{k}\right),$ such as growth and coercivity conditions, or the AmbrosettiRabinowitz condition.
Remark 4.
Our theory applies in particular to a single functional $E$ defined on a product space ${H}_{1}\times {H}_{2},$ when we can take either
 (1^{0})

${E}_{1}={E}_{2}=E;$or
 (2^{0})

${E}_{1}=E$ and ${E}_{2}=E.$
The results for case (2^{0}) will be in some sense dual to those for case (1^{0}). Thus, one can produce critical points $({u}_{1}^{\ast},{u}_{2}^{\ast})$ of $E,$ with one of the properties:
$E({u}_{1}^{\ast},{u}_{2}^{\ast})$  $=$  $\mathrm{min}E(\cdot ,{u}_{2}^{\ast})=\mathrm{max}E({u}_{1}^{\ast},\cdot );$  
$E({u}_{1}^{\ast},{u}_{2}^{\ast})$  $=$  $\mathrm{min}E(\cdot ,{u}_{2}^{\ast})=\underset{\mu \in {\mathrm{\Gamma}}_{2}}{sup}\underset{q\in {Q}_{2}}{\mathrm{min}}E({u}_{1}^{\ast},\mu \left(q\right));$  
$E({u}_{1}^{\ast},{u}_{2}^{\ast})$  $=$  $\underset{\mu \in {\mathrm{\Gamma}}_{1}}{inf}\underset{q\in {Q}_{1}}{\mathrm{max}}E(\mu \left(q\right),{u}_{2}^{\ast})=\mathrm{max}E({u}_{1}^{\ast},\cdot );$  
$E({u}_{1}^{\ast},{u}_{2}^{\ast})$  $=$  $\underset{\mu \in {\mathrm{\Gamma}}_{1}}{inf}\underset{q\in {Q}_{1}}{\mathrm{max}}E(\mu \left(q\right),{u}_{2}^{\ast})=\underset{\mu \in {\mathrm{\Gamma}}_{2}}{sup}\underset{q\in {Q}_{2}}{\mathrm{min}}E({u}_{1}^{\ast},\mu \left(q\right)).$ 
3 Application
We apply the results from the previous section to the Dirichlet problem
$$\{\begin{array}{cc}\mathrm{\Delta}{v}_{1}={\nabla}_{{v}_{1}}F({v}_{1},{w}_{1},{v}_{2},{w}_{2})\hfill & \\ \mathrm{\Delta}{w}_{1}={\nabla}_{{w}_{1}}F({v}_{1},{w}_{1},{v}_{2},{w}_{2})\hfill & \\ \mathrm{\Delta}{v}_{2}={\nabla}_{{v}_{2}}G({v}_{1},{w}_{1},{v}_{2},{w}_{2})\phantom{\rule{1em}{0ex}}\hfill & \\ \mathrm{\Delta}{w}_{2}={\nabla}_{{w}_{2}}G({v}_{1},{w}_{1},{v}_{2},{w}_{2})\phantom{\rule{1.5em}{0ex}}\text{on}\mathrm{\Omega}\hfill & \\ {{v}_{1}}_{\partial \mathrm{\Omega}}={{w}_{1}}_{\partial \mathrm{\Omega}}={{v}_{2}}_{\partial \mathrm{\Omega}}={{w}_{2}}_{\partial \mathrm{\Omega}}=0,\hfill & \end{array}$$  (23) 
where $\mathrm{\Omega}$ is a bounded open set from ${\mathbb{R}}^{n}$ $(n\ge 3)$. These kinds of problems are widely recognized in the literature and they model realworld processes, such as stationary diffusion or wave propagation.
Throughout the section, the symbol $\cdot $ is used to denote the Euclidean norm in ${\mathbb{R}}^{2}$. We assume the following behavior of potentials $F$ and $G:$

(H1)
$F,G:{\mathbb{R}}^{4}\to \mathbb{R}$ are of ${C}^{1}$ class and satisfy
$$F(0,{x}_{2})=0\phantom{\rule{1.5em}{0ex}}\text{and}G({x}_{1},0)=0,$$ for all ${x}_{1},{x}_{2}\in {\mathbb{R}}^{2},$ and the growth conditions
$\leftF({x}_{1},{x}_{2})\right$ $\le $ ${C}_{F}\left({\left{x}_{1}\right}^{p}+1\right),$ (24) $\leftG({x}_{1},{x}_{2})\right$ $\le $ ${C}_{G}\left({\left{x}_{2}\right}^{p}+1\right),$ for all ${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}$ and some positive constants ${C}_{F},{C}_{G},$ where $2\le p\le {2}^{\ast}=\frac{2n}{n2}.$
Here we take ${H}_{1}={H}_{2}:=$ ${\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}={H}_{0}^{1}(\mathrm{\Omega})\times {H}_{0}^{1}(\mathrm{\Omega})$ endowed with the inner product
$${(u,\overline{u})}_{{H}_{0}^{1}\times {H}_{0}^{1}}={(v,\overline{v})}_{{H}_{0}^{1}}+{(w,\overline{w})}_{{H}_{0}^{1}},$$ 
and the corresponding norm
$${u}_{{H}_{0}^{1}\times {H}_{0}^{1}}={\left({v}_{{H}_{0}^{1}}^{2}+{w}_{{H}_{0}^{1}}^{2}\right)}^{1/2},$$ 
for $u=(v,w),\overline{u}=(\overline{v},\overline{w}).$
The defining characteristic of the considered system (23) is that the first two and last two equations, coupled together, allow for a variational formulation given by the energy functionals ${E}_{1},{E}_{2}:{\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}\times {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}\to \mathbb{R},$
${E}_{1}({u}_{1},{u}_{2})=\frac{1}{2}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle {\int}_{\mathrm{\Omega}}}F({u}_{1},{u}_{2}),$  
${E}_{2}({u}_{1},{u}_{2})=\frac{1}{2}{{u}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle {\int}_{\mathrm{\Omega}}}G({u}_{1},{u}_{2}),$ 
where ${u}_{1}=({v}_{1},{w}_{1}),{u}_{2}=({v}_{2},{w}_{2})\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$.
We are interested in a weak solution $({u}_{1}^{\ast},{u}_{2}^{\ast})$ of (23) such that $({u}_{1}^{\ast},{u}_{2}^{\ast})$ is a mountain passmin point for the functionals ${E}_{1},{E}_{2}$, i.e., ${u}_{1}^{\ast}$ is a mountain pass type critical point for ${E}_{1}(\cdot ,{u}_{2}^{\ast})$ and ${u}_{2}^{\ast}$ is a minimizer for ${E}_{2}({u}_{1}^{\ast},\cdot )$.
Letting
${f}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})={\nabla}_{{y}_{1}}F({y}_{1},{z}_{1},{y}_{2},{z}_{2}),$  
${f}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})={\nabla}_{{z}_{1}}F({y}_{1},{z}_{1},{y}_{2},{z}_{2}),$  
${g}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})={\nabla}_{{y}_{2}}G({y}_{1},{z}_{1},{y}_{2},{z}_{2}),$  
${g}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})={\nabla}_{{z}_{2}}G({y}_{1},{z}_{1},{y}_{2},{z}_{2}),$ 
the identification of ${H}^{1}(\mathrm{\Omega})$ with ${H}_{0}^{1}(\mathrm{\Omega})$ via $\mathrm{\Delta}$ yields to the representation
${E}_{11}({u}_{1},{u}_{2})$  $=$  ${u}_{1}({(\mathrm{\Delta})}^{1}{f}_{1}({u}_{1},{u}_{2}),{(\mathrm{\Delta})}^{1}{f}_{2}({u}_{1},{u}_{2})),$  
${E}_{22}({u}_{1},{u}_{2})$  $=$  ${u}_{2}({(\mathrm{\Delta})}^{1}{g}_{1}({u}_{1},{u}_{2}),{(\mathrm{\Delta})}^{1}{g}_{2}({u}_{1},{u}_{2})).$ 
Note that under the growth conditions (24), the Nemytskii’s operators
$${\mathcal{N}}_{{f}_{i}}({u}_{1},{u}_{2})(x):={f}_{i}({u}_{1}(x),{u}_{2}(x)),{\mathcal{N}}_{{g}_{i}}({u}_{1},{u}_{2})(x):={g}_{i}({u}_{1}(x),{u}_{2}(x)),$$ 
(i=1,2), are well defined from ${\left({L}^{{2}^{\ast}}(\mathrm{\Omega})\right)}^{4}$ to ${\left({L}^{{\left({2}^{\ast}\right)}^{\prime}}(\mathrm{\Omega})\right)}^{2}$, continuous and bounded (map bounded sets into bounded sets). Consequently, the operators
${N}_{1}({u}_{1},{u}_{2})=({(\mathrm{\Delta})}^{1}{f}_{1}({u}_{1},{u}_{2})),{(\mathrm{\Delta})}^{1}{f}_{2}({u}_{1},{u}_{2})))$  
${N}_{2}({u}_{1},{u}_{2})=({(\mathrm{\Delta})}^{1}{g}_{1}({u}_{1},{u}_{2})),{(\mathrm{\Delta})}^{1}{g}_{2}({u}_{1},{u}_{2})))$ 
are welldefined and continuous from ${\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{4}$ to ${\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$.
Let ${\lambda}_{1}$ be the first eigenvalue of the Dirichlet problem $\mathrm{\Delta}u=\lambda v$ in $\mathrm{\Omega},v=0$ on $\partial \mathrm{\Omega}$ (see, e.g., (14)). Our next hypothesis is a crossing condition of the first eigenvalue which has been used in the literature (see, e.g., (6), (12) and (25)).

(H2)
The inequalities
$$ hold for all ${y}_{1}\in \mathbb{R}$ and uniformly with respect to ${x}_{2}\in {\mathbb{R}}^{2}.$
From (24) and (H2), there are $\mu ,\tau $ with $$ and ${C}_{\mu},{C}_{\tau}>0$ such that
$$\frac{\tau}{2}{y}_{1}{}_{}{}^{2}{C}_{\tau}\le F(({y}_{1},0),{x}_{2})\text{and}F({x}_{1},{x}_{2})\le \frac{\mu}{2}{x}_{1}{}^{2}+{C}_{\mu}{x}_{1}{}^{p},$$  (25) 
for all ${y}_{1}\in \mathbb{R}$ and ${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}.$
One can see that the first inequality of (25) guarantees
${E}_{1}((\alpha {\varphi}_{1},0),{u}_{2})$  $=\frac{1}{2}{\left(\alpha {\varphi}_{1},0)\right}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle {\int}_{\mathrm{\Omega}}}F((\alpha {\varphi}_{1},0),{u}_{2})$  (26)  
$\le \frac{1}{2}{\alpha}^{2}{{\varphi}_{1}}_{{H}_{0}^{1}}^{2}\frac{1}{2}\tau {\alpha}^{2}{{\varphi}_{1}}_{{L}^{2}}^{2}+{C}_{\tau}\text{meas}(\mathrm{\Omega})$  
$=\frac{1}{2}\left(1\frac{\tau}{{\lambda}_{1}}\right){\alpha}^{2}+{C}_{\tau}\text{meas}(\mathrm{\Omega})\to \mathrm{\infty},\text{as}\alpha \to \mathrm{\infty},$ 
uniformly with respect to ${u}_{2}\in {\left({H}_{0}^{1}\left(\mathrm{\Omega}\right)\right)}^{2}$, whilst the second one implies
${E}_{1}({u}_{1},{u}_{2})$  $=\frac{1}{2}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle {\int}_{\mathrm{\Omega}}}F({u}_{1},{u}_{2})$  
$\ge \frac{1}{2}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}\frac{\mu}{2}{\displaystyle {\int}_{\mathrm{\Omega}}}{{u}_{1}(x)}^{2}{C}_{\mu}{\displaystyle {\int}_{\mathrm{\Omega}}}{{u}_{1}(x)}^{p}\mathit{d}x$  
$\ge \frac{1}{2}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}\frac{\mu}{2{\lambda}_{1}}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{C}_{\mu}^{\prime}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{p}$  
$=\left(\frac{1}{2}\frac{\mu}{2{\lambda}_{1}}\right){{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{C}_{\theta}^{\prime}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{p}.$ 
Given that $\frac{1}{2}\frac{\mu}{2{\lambda}_{1}}>0$, there exists ${r}_{0}^{\prime}>0$, sufficiently small and independent of ${u}_{2}$, and a constant $c>0,$ such that
$${E}_{1}({u}_{1},{u}_{2})\ge c>0\phantom{\rule{1.5em}{0ex}}\text{whenever}{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}={r}_{0}^{\prime}.$$  (27) 
Based on (26), we can choose ${\alpha}_{0}>{r}_{0}^{\prime}$ such that
$$  (28) 
In addition, one has
$${E}_{1}((0,0),{u}_{2})=0.$$  (29) 
Now, if we consider in ${\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$ the sets
${A}_{1}=\{{u}_{1}\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}:{{u}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}={r}_{0}^{\prime}\},$  
${Q}_{1}=\{s({\varphi}_{1},0)\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}:\mathrm{\hspace{0.17em}0}\le s\le {\alpha}_{0}\},$  
${B}_{1}=\left\{((0,0),({s}_{0}{\varphi}_{1},0))\right\},$ 
then, from (27), (28), (29), we see that ${A}_{1}$ links ${B}_{1}$ via ${Q}_{1}$, and moreover
$$\underset{{A}_{1}}{inf}{E}_{1}(\cdot ,{u}_{2})\ge c>\underset{{B}_{1}}{sup}{E}_{1}(\cdot ,{u}_{2}),$$ 
for all ${u}_{2}\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$, i.e., $$
Also take
$${A}_{2}={\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2},{B}_{2}=\mathrm{\varnothing}\phantom{\rule{1.5em}{0ex}}\text{and}{Q}_{2}=\{(0,0)\},$$ 
which corresponds to the trivial linking. Furthermore, in order to have $$ equivalently $$ the functional ${E}_{2}(\cdot ,{u}_{2})$ must be bounded from below uniformly with respect to ${u}_{1}$. This requirement can be satisfied by the imposition of the following unilateral growth condition on $G$:

(H3)
There exists $$ with
$$G({x}_{1},{x}_{2})\le \frac{\sigma}{2}{\left{x}_{2}\right}^{2}+C,$$ (30) for all ${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}.$
As a result of Theorem 4, it can be inferred that there exist two sequences, $\left({u}_{1}^{k}\right),\left({u}_{2}^{k}\right)$ which satisfies (3) and (4).
In what follows, we will establish sufficient conditions for the convergence of the sequences $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ previously constructed. With reference to Theorem 6, we consider in this case, the operator $L=({L}_{1},{L}_{2}),$ here linear, with${L}_{1},{L}_{2}:{\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}\to {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$ defined as
$${L}_{1}({v}_{1},{w}_{1})={L}_{1}\left({u}_{1}\right)=\beta ({v}_{1}{w}_{1},{v}_{1}{w}_{1}),{L}_{2}({v}_{2},{w}_{2})={L}_{2}\left({u}_{2}\right)={u}_{2},$$  (31) 
for ${u}_{1}=({v}_{1},{w}_{1}),{u}_{2}=({v}_{2},{w}_{2})\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$ and some $\beta >0.$ Thus, correspondingly, one has
${N}_{1}({u}_{1},{u}_{2})$  
$={u}_{1}{L}_{1}\left({E}_{11}({u}_{1},{u}_{2})\right)$  
$={u}_{1}{L}_{1}{u}_{1}+{L}_{1}({(\mathrm{\Delta})}^{1}{f}_{1}({u}_{1},{u}_{2})),{(\mathrm{\Delta})}^{1}{f}_{2}({u}_{1},{u}_{2})))$  
$=((1\beta ){v}_{1}+\beta {w}_{1},(1\beta ){w}_{1}\beta {v}_{1})$  
$+\beta ({(\mathrm{\Delta})}^{1}\left({f}_{1}({u}_{1},{u}_{2}){f}_{2}({u}_{1},{u}_{2})\right),{(\mathrm{\Delta})}^{1}\left({f}_{1}({u}_{1},{u}_{2}){f}_{2}({u}_{1},{u}_{2})\right))$ 
and
${N}_{2}({u}_{1},{u}_{2})$  $={u}_{2}{L}_{2}\left({E}_{22}({u}_{1},{u}_{2})\right)$  
$={u}_{2}{L}_{2}{u}_{2}+{L}_{2}({(\mathrm{\Delta})}^{1}{g}_{1}({u}_{1},{u}_{2})),{(\mathrm{\Delta})}^{1}{g}_{2}({u}_{1},{u}_{2})))$  
$=({(\mathrm{\Delta})}^{1}{g}_{1}({u}_{1},{u}_{2}),{(\mathrm{\Delta})}^{1}{g}_{2}({u}_{1},{u}_{2}))\phantom{\rule{5.22222222222224em}{0ex}}$ 
Next we introduce some monotonicity conditions on the functions $\stackrel{~}{f}:={f}_{1}{f}_{2},$ ${g}_{1}$ and ${g}_{2}$ which are involved in the above expressions of ${N}_{1}$ and ${N}_{2}.$
It is worth noting that these conditions are applied to differences and do not impose restrictions on $F$ of being quadratic, as is the case with $G$ according to (H3). Examples 1 and 2 that follow support this assertion.

(H4)
There are nonnegative numbers ${m}_{ij}$ ($i,j=1,4$) such that
$\left(\stackrel{~}{f}({x}_{1},{x}_{2})\stackrel{~}{f}({\overline{x}}_{1},{\overline{x}}_{2})\right)\left({y}_{1}{\overline{y}}_{1}\right)$ $\le $ ${y}_{1}{\overline{y}}_{1}\left({m}_{11}{y}_{1}{\overline{y}}_{1}+{m}_{12}{z}_{1}{\overline{z}}_{1}+{m}_{13}{y}_{2}{\overline{y}}_{2}+{m}_{14}{z}_{2}{\overline{z}}_{2}\right),$ $\left(\stackrel{~}{f}({x}_{1},{x}_{2})\stackrel{~}{f}({\overline{x}}_{1},{\overline{x}}_{2})\right)\left({z}_{1}{\overline{z}}_{1}\right)$ $\le $ ${z}_{1}{\overline{z}}_{1}\left({m}_{21}{y}_{1}{\overline{y}}_{1}+{m}_{22}{z}_{1}{\overline{z}}_{1}+{m}_{23}{y}_{2}{\overline{y}}_{2}+{m}_{24}{z}_{2}{\overline{z}}_{2}\right),$ $\left({g}_{1}({x}_{1},{x}_{2}){g}_{1}({\overline{x}}_{1},{\overline{x}}_{2})\right)\left({y}_{2}{\overline{y}}_{2}\right)$ $\le $ ${y}_{2}{\overline{y}}_{2}({m}_{31}{y}_{1}{\overline{y}}_{1}+{m}_{32}{z}_{1}{\overline{z}}_{1}+{m}_{33}{y}_{2}{\overline{y}}_{2}+{m}_{34}{z}_{2}{\overline{z}}_{2}),$ $\left({g}_{2}({x}_{1},{x}_{2}){g}_{2}({\overline{x}}_{1},{\overline{x}}_{2})\right)\left({z}_{2}{\overline{z}}_{2}\right)$ $\le $ ${z}_{2}{\overline{z}}_{2}\left({m}_{41}{y}_{1}{\overline{y}}_{1}+{m}_{42}{z}_{1}{\overline{z}}_{1}+{m}_{43}{y}_{2}{\overline{y}}_{2}+{m}_{44}{z}_{2}{\overline{z}}_{2}\right),$ for all ${x}_{1}=({y}_{1},{z}_{1}),{\overline{x}}_{1}=({\overline{y}}_{1},{\overline{z}}_{1}),{x}_{2}=({y}_{2},{z}_{2}),{\overline{x}}_{2}=({\overline{y}}_{2},{\overline{z}}_{2})\in {\mathbb{R}}^{2}$.
Under assumption (H4), the operators ${N}_{1},{N}_{2}$ satisfy the monotonicity conditions (14) and (15) with the following coefficients:
${a}_{11}$  $=1\beta +\frac{\beta}{{\lambda}_{1}}\mathrm{max}\{{m}_{11},{m}_{22}\}+\frac{\beta}{2{\lambda}_{1}}\left({m}_{12}+{m}_{21}\right),$  (33)  
${a}_{12}$  $=\frac{\beta}{{\lambda}_{1}}\mathrm{max}\{\sqrt{{m}_{13}^{2}+{m}_{23}^{2}},\sqrt{{m}_{14}^{2}+{m}_{24}^{2}}\},$  (34)  
${a}_{21}$  $={\displaystyle \frac{1}{{\lambda}_{1}}}\mathrm{max}\{\sqrt{{m}_{31}^{2}+{m}_{32}^{2}},\sqrt{{m}_{41}^{2}+{m}_{42}^{2}}\},$  (35)  
${a}_{22}$  $={\displaystyle \frac{{m}_{34}+{m}_{43}}{2{\lambda}_{1}}}+\mathrm{max}\{{m}_{33},{m}_{44}\}.$  (36) 
Indeed, for any ${u}_{1}=({v}_{1},{w}_{1}),{u}_{2},{\overline{u}}_{1},{\overline{u}}_{2}\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2},$ we have
${({N}_{1}({u}_{1},{u}_{2}){N}_{1}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{1}{\overline{u}}_{1})}_{{H}_{0}^{1}\times {H}_{0}^{1}}$  
$=\left(1\beta \right){\left{u}_{1}{\overline{u}}_{1}\right}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}+\beta {(\stackrel{~}{f}({u}_{1},{u}_{2})\stackrel{~}{f}({\overline{u}}_{1},{\overline{u}}_{2}),{v}_{1}{\overline{v}}_{1})}_{{L}^{2}}$  
$+\beta {(\stackrel{~}{f}({u}_{1},{u}_{2})\stackrel{~}{f}({\overline{u}}_{1},{\overline{u}}_{2}),{w}_{1}{\overline{w}}_{1})}_{{L}^{2}}.$ 
Using (H4) and the well known inequality ${v}_{{L}^{2}}{w}_{{L}^{2}}\le \frac{1}{2}\left({v}_{{L}^{2}}^{2}+{w}_{{L}^{2}}^{2}\right),$ we obtain
${({N}_{1}({u}_{1},{u}_{2}){N}_{1}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{1}{\overline{u}}_{1})}_{{H}_{0}^{1}\times {H}_{0}^{1}}$  
$\le \left(1\beta \right)\left({{v}_{1}{\overline{u}}_{1}}_{{H}_{0}^{1}}^{2}+{{w}_{1}{\overline{w}}_{1}}_{{H}_{0}^{1}}^{2}\right)$  
$+\beta {m}_{11}{{v}_{1}{\overline{v}}_{1}}_{{L}^{2}}^{2}+\beta {m}_{12}{\left{v}_{1}{\overline{v}}_{1}\right}_{{L}^{2}}{\left{w}_{1}{\overline{w}}_{1}\right}_{{L}^{2}}$  
$+\beta {m}_{22}{{w}_{1}{\overline{w}}_{1}}_{{L}^{2}}^{2}+\beta {m}_{21}{\left{v}_{1}{\overline{v}}_{1}\right}_{{L}^{2}}{\left{w}_{1}{\overline{w}}_{1}\right}_{{L}^{2}}$  
$+\beta {m}_{13}{\left{v}_{1}{\overline{v}}_{1}\right}_{{L}^{2}}{\left{v}_{2}{\overline{v}}_{2}\right}_{{L}^{2}}+{m}_{14}{\left{v}_{1}{\overline{v}}_{1}\right}_{{L}^{2}}{\left{w}_{2}{\overline{w}}_{2}\right}_{{L}^{2}}$  
$+\beta {m}_{23}{\left{w}_{1}{\overline{w}}_{1}\right}_{{L}^{2}}{\left{v}_{2}{\overline{v}}_{2}\right}_{{L}^{2}}+{m}_{24}{\left{w}_{1}{\overline{w}}_{1}\right}_{{L}^{2}}{\left{w}_{2}{\overline{w}}_{2}\right}_{{L}^{2}}.$ 
As both ${{v}_{1}{\overline{v}}_{1}}_{{L}^{2}}$ and ${\left{w}_{1}{\overline{w}}_{1}\right}_{{L}^{2}}$ are less or equal to ${{u}_{1}{\overline{u}}_{1}}_{{L}^{2}\times {L}^{2}},$ from Poincaré’s inequality ${v}_{{L}^{2}}^{2}\le \frac{1}{{\lambda}_{1}}{v}_{{H}_{0}^{1}}^{2},$ we infer that
${({N}_{1}({u}_{1},{u}_{2}){N}_{1}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{1}{\overline{u}}_{1})}_{{H}_{0}^{1}\times {H}_{0}^{1}}$  
$\le {a}_{11}{\left{u}_{1}{\overline{u}}_{1}\right}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}+{a}_{12}{{u}_{1}{\overline{u}}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}{{u}_{2}{\overline{u}}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}.$ 
Similarly, we have
${({N}_{2}({u}_{1},{u}_{2}){N}_{2}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{2}{\overline{u}}_{2})}_{{H}_{0}^{1}\times {H}_{0}^{1}}$  
$\le {m}_{33}{{v}_{2}{\overline{v}}_{2}}_{{H}_{0}^{1}}^{2}+{m}_{44}{{w}_{2}{\overline{w}}_{2}}_{{H}_{0}^{1}}^{2}$  
$+\left({m}_{34}+{m}_{43}\right){{v}_{2}{\overline{v}}_{2}}_{{L}^{2}}{{w}_{2}{\overline{w}}_{2}}_{{L}^{2}}$  
$+{{v}_{1}{\overline{v}}_{1}}_{{L}^{2}}\left({m}_{31}{{v}_{1}{\overline{v}}_{1}}_{{L}^{2}}+{m}_{32}{{w}_{1}{\overline{w}}_{2}}_{{L}^{2}}\right)$  
$+{{v}_{2}{\overline{v}}_{2}}_{{L}^{2}}\left({m}_{41}{{v}_{1}{\overline{v}}_{1}}_{{L}^{2}}+{m}_{42}{{w}_{1}{\overline{w}}_{2}}_{{L}^{2}}\right),$ 
which after further computation gives
${({N}_{2}({u}_{1},{u}_{2}){N}_{2}({\overline{u}}_{1},{\overline{u}}_{2}),{u}_{2}{\overline{u}}_{2})}_{{H}_{0}^{1}\times {H}_{0}^{1}}$  
$\le {a}_{22}{{u}_{2}{\overline{u}}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}+{a}_{21}{{u}_{1}{\overline{u}}_{1}}_{{H}_{0}^{1}\times {H}_{0}^{1}}{{u}_{2}{\overline{u}}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}.$ 
Now it is clear that the first two conditions outlined in Theorem 6 are satisfied provided that

(H5)
The matrix $M:={[{a}_{ij}]}_{1\le i,j\le 2}$ is convergent to zero.
It remains to show that the sequence $\left({u}_{2}^{k}\right)$ is bounded. To this aim we use Theorem 7 (a). First, since $G(\cdot ,0)=0,$ we clearly have ${E}_{2}({u}_{1},0)=0,$ for any ${u}_{1}\in {({H}_{0}^{1}(\mathrm{\Omega}))}^{2}.$ Next, the growth condition (30) on $G$ gives
${E}_{2}({u}_{1},{u}_{2})$  $=\frac{1}{2}{{u}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle {\int}_{\mathrm{\Omega}}}G({u}_{1},{u}_{2})$  
$\ge \frac{1}{2}{{u}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}{\displaystyle \frac{\sigma}{2}}{\left{u}_{2}\right}_{{L}^{2}\left(\mathrm{\Omega}\right)\times {L}^{2}\left(\mathrm{\Omega}\right)}^{2}C\text{meas}\left(\mathrm{\Omega}\right)$  
$\ge \left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sigma}{2{\lambda}_{1}}}\right){{u}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}^{2}C\text{meas}\left(\mathrm{\Omega}\right)\to \mathrm{\infty},$ 
$\text{as}{{u}_{2}}_{{H}_{0}^{1}\times {H}_{0}^{1}}\to \mathrm{\infty}$, uniformly with respect to ${u}_{1}.$ Therefore, as all conditions outlined in Theorem 6 are fulfilled, it can be deduced that the sequences $\left({u}_{1}^{k}\right)$ and $\left({u}_{2}^{k}\right)$ are convergent in ${\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}.$
Thus, based on Theorem 4, we can state the following theorem.
Theorem 8.
Assume that (H1)(H5) hold. Then problem (23) has a mountain passmin solution, i.e., there is a solution $({u}_{1}^{\ast},{u}_{2}^{\ast})\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}\times {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$ such that ${u}_{1}^{\ast}$ is a mountain pass type critical point of the functional ${E}_{1}(\cdot ,{u}_{2}^{\ast})$ and ${u}_{2}^{\ast}$ is a minimizer of the functional ${E}_{2}({u}_{1}^{\ast},\cdot )$.
To attain a mountain passmountain pass solution, we follow a similar approach as in Theorem 8, with some important clarifications. Firstly, it is necessary to impose the conditions from (H2) on both $F$ and $G$ (denote this condition with (H2)’) in order to guarantee that both nontrivial linkings are fulfilled. Furthermore, it is easy to see that imposing (H3) with $G$ instead of $G$ (denote this condition with (H3)’), we guarantee the boundedness of the sequence ${u}_{2}^{k}$, as indicated by Theorem 7(b).
Secondly, we must take into account a different operator ${L}_{2}$ than the identity, since, as noted in Remark 2, selecting ${L}_{2}=\text{Id}$ results in a minimum point. For simplicity, we take ${L}_{2}:={L}_{1}$, where ${L}_{1}$ is defined in (31). Thus, the alteration in condition (H4) is that we require monotonicity for $\stackrel{~}{g}$, instead of ${g}_{1}$ and ${g}_{2}$ (denote this condition with (H4)’), with $\stackrel{~}{g}$ defined as $\stackrel{~}{g}={g}_{1}{g}_{2}$. Changing the operator ${L}_{2}$ results in revising the coefficients ${a}_{21}$ and ${a}_{22}$ as outlined in equations $(\text{35})$ and $(\text{36})$, with ${a}_{21}$ being equivalent to ${a}_{12}$ and ${a}_{22}$ being equivalent to ${a}_{11}$, as per equations $(\text{33})$ and $(\text{34})$.
Therefore, we can state the following result.
Theorem 9.
Assume that (H1), (H2)’(H4)’, (H5) holds true. Then problem (23) has a mountain passmountain pass solution, i.e., there is a solution $({u}_{1}^{\ast},{u}_{2}^{\ast})\in {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}\times {\left({H}_{0}^{1}(\mathrm{\Omega})\right)}^{2}$ such that ${u}_{1}^{\ast}$ is a mountain pass type critical point of the functional ${E}_{1}(\cdot ,{u}_{2}^{\ast})$ and ${u}_{2}^{\ast}$ mountain pass type critical point of the functional ${E}_{2}({u}_{1}^{\ast},\cdot )$.
Example 1. Consider the Dirichlet problem
$$\{\begin{array}{cc}\mathrm{\Delta}{v}_{1}=a{({v}_{1}+{w}_{1})}^{3}+\stackrel{~}{a}{v}_{1}+a({v}_{1}+{w}_{1})\frac{1}{{v}_{2}^{2}+{w}_{2}^{2}+1}\hfill & \\ \mathrm{\Delta}{w}_{1}=a{({v}_{1}+{w}_{1})}^{3}\stackrel{~}{a}{w}_{1}+a({v}_{1}+{w}_{1})\frac{1}{{v}_{2}^{2}+{w}_{2}^{2}+1}\hfill & \\ \mathrm{\Delta}{v}_{2}=b{v}_{2}+\frac{1}{{v}_{1}^{2}+{c}^{2}}\hfill & \\ \mathrm{\Delta}{w}_{2}=b{w}_{2}+\frac{1}{{v}_{2}^{2}+{c}^{2}}\hfill & \end{array}$$  (37) 
We apply Theorem 8, where
$$  
$F({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\frac{a}{4}{({y}_{1}+{z}_{1})}^{4}+\frac{\stackrel{~}{a}}{2}\left({y}_{1}^{2}{z}_{1}^{2}\right)+\frac{a}{2}{({y}_{1}+{z}_{1})}^{2}{\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
$G({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\frac{b}{2}\left({y}_{2}^{2}+{z}_{2}^{2}\right)+{\displaystyle \frac{{y}_{2}}{{y}_{1}^{2}+{c}^{2}}}+{\displaystyle \frac{{z}_{2}}{{z}_{1}^{2}+{c}^{2}}}.$ 
One can easily see that the absolute value of $F({x}_{1},{x}_{2})$ (${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}$) is upperbounded by a fourthdegree polynomial in $\left{x}_{1}\right$ and
$$\leftG({y}_{1},{z}_{1},{y}_{2},{z}_{2})\right\le \left(\frac{b}{2}+\frac{2}{c}\right){\left({y}_{2},{z}_{2})\right}^{2}+\frac{2}{c}.$$ 
Thus condition (H1) is satisfied. Also, condition (H3) holds as $$
Verification of the condition (H2). Since $\frac{{({y}_{1}+{z}_{1})}^{4}}{{y}_{1}^{2}+{z}_{1}^{2}}\to 0$ provided ${y}_{1}+{z}_{1}\to 0,$ simple computations yields
$$ 
On the other hand,
$\underset{{y}_{1}\to \mathrm{\infty}}{lim}{\displaystyle \frac{F(({y}_{1},0),{x}_{2})}{{y}_{1}{}^{2}}}$  $=$  $\underset{{y}_{1}\to \mathrm{\infty}}{lim}{\displaystyle \frac{\frac{a}{4}{y}_{1}^{4}+\frac{\stackrel{~}{a}}{2}{y}_{1}^{2}+\frac{a}{2}{y}_{1}^{2}\frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}}{{y}_{1}{}^{2}}}$  
$\ge $  $\underset{{y}_{1}\to \mathrm{\infty}}{lim}\frac{a}{4}{y}_{1}^{2}=\mathrm{\infty},$ 
uniformly with respect to ${x}_{2}=({y}_{2},{z}_{2})\in {\mathbb{R}}^{2}$. Thus (H2) holds.
Verification of the condition (H4). First note that
${f}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{1}+{z}_{1})}^{3}+\stackrel{~}{a}{y}_{1}+a({y}_{1}+{z}_{1}){\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
${f}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{1}+{z}_{1})}^{3}\stackrel{~}{a}{z}_{1}+a({y}_{1}+{z}_{1}){\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
${g}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=b{y}_{2}+{\displaystyle \frac{1}{{y}_{1}^{2}+{c}^{2}}},$  
${g}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=b{z}_{2}+{\displaystyle \frac{1}{{z}_{1}^{2}+{c}^{2}}},$ 
which clearly gives
$$\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\stackrel{~}{a}{y}_{1}+\stackrel{~}{a}{z}_{1}.$$ 
Note that the function $h:\mathbb{R}\to \mathbb{R}$ defined as $h(x)=\frac{1}{{x}^{2}+{c}^{2}}$ is Lipschitz continuous, with a Lipschitz constant not greater than $\frac{1}{c},$ provided that $c\ge 1$, i.e.,
$$\left\frac{1}{{x}^{2}+{c}^{2}}\frac{1}{{\overline{x}}^{2}+{c}^{2}}\right\le \frac{1}{c}x\overline{x},\text{for all}x,\overline{x}\in \mathbb{R}.\text{}$$  (38) 
From the linearity of $\stackrel{~}{f}$ and the Lipschitz property (38), it follows that
$\left(\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{f}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({y}_{1}{\overline{y}}_{1})$  
$\le \stackrel{~}{a}{{y}_{1}{\overline{y}}_{1}}^{2}+\stackrel{~}{a}\left{y}_{1}{\overline{y}}_{1}\right{z}_{1}{\overline{z}}_{1},$  
$\left(\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{f}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({z}_{1}{\overline{z}}_{1})$  
$\le \stackrel{~}{a}{{z}_{1}{\overline{z}}_{1}}^{2}+\stackrel{~}{a}\left{y}_{1}{\overline{y}}_{1}\right{z}_{1}{\overline{z}}_{1},$  
$\left({g}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2}){g}_{1}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({y}_{2}{\overline{y}}_{2})$  
$\le b{{y}_{2}{\overline{y}}_{2}}^{2}+\frac{1}{c}{y}_{2}{\overline{y}}_{2}{y}_{1}{\overline{y}}_{1},$  
$\left({g}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})g({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({z}_{2}{\overline{z}}_{2})$  
$\le b{{z}_{2}{\overline{z}}_{2}}^{2}+\frac{1}{c}{z}_{1}{\overline{z}}_{1}{z}_{2}{\overline{z}}_{2}.$ 
Hence the monotonicity conditions ((H4)) hold with
${m}_{11}$  $=$  $\stackrel{~}{a},{m}_{12}=\stackrel{~}{a},{m}_{13}=0,{m}_{14}=0,$  
${m}_{21}$  $=$  $\stackrel{~}{a},{m}_{22}=\stackrel{~}{a},{m}_{23}=0,{m}_{24}=0,$  
${m}_{31}$  $=$  $\frac{1}{c}},{m}_{32}=0,{m}_{33}=b,{m}_{34}=0,$  
${m}_{41}$  $=$  $0,{m}_{42}={\displaystyle \frac{1}{c}},{m}_{43}=0,{m}_{44}=b.$ 
Verification of the condition (H5). Simple computations yield
$$M=\left[\begin{array}{cc}1\beta \left(12\frac{\stackrel{~}{a}}{{\lambda}_{1}}\right)& 0\\ \frac{1}{c{\lambda}_{1}}& b\end{array}\right].$$ 
Since $$ and $12\frac{\stackrel{~}{a}}{{\lambda}_{1}}>0,$ we can choose $\beta >0$ in (31) sufficiently small that the matrix $M$ is convergent to zero.
Thus all the hypothesis of Theorem 8 are satisfied and problem (37) has a solution $({v}_{1}^{\ast},{w}_{1}^{\ast},{v}_{2}^{\ast},{w}_{2}^{\ast}),$ where if ${u}_{1}^{\ast}:=$ $({v}_{1}^{\ast},{w}_{1}^{\ast})$ and ${u}_{2}^{\ast}:=({v}_{2}^{\ast},{w}_{2}^{\ast}),$ one has that ${u}_{1}^{\ast}$ is a mountain pass type critical point of the energy functional ${E}_{1}(\cdot ,{u}_{2}^{\ast}),$ and ${u}_{2}^{\ast}$ is a minimizer of the energy functional ${E}_{2}({u}_{1}^{\ast},\cdot ).$
Example 2. Consider the Dirichlet problem
$$\{\begin{array}{cc}\mathrm{\Delta}{v}_{1}=a{({v}_{1}+{w}_{1})}^{3}+\stackrel{~}{a}{v}_{1}+a({v}_{1}+{w}_{1})\frac{1}{{v}_{2}^{2}+{w}_{2}^{2}+1}\hfill & \\ \mathrm{\Delta}{w}_{1}=a{({v}_{1}+{w}_{1})}^{3}\stackrel{~}{a}{w}_{1}+a({v}_{1}+{w}_{1})\frac{1}{{v}_{2}^{2}+{w}_{2}^{2}+1}\hfill & \\ \mathrm{\Delta}{v}_{2}=a{({v}_{2}+{w}_{2})}^{3}+\stackrel{~}{a}{v}_{2}+a({v}_{2}+{w}_{2})\frac{1}{{v}_{1}^{2}+{w}_{1}^{2}+1}\hfill & \\ \mathrm{\Delta}{w}_{2}=a{({v}_{2}+{w}_{2})}^{3}\stackrel{~}{a}{w}_{2}+a({v}_{2}+{w}_{2})\frac{1}{{v}_{1}^{2}+{w}_{1}^{2}+1}\hfill & \end{array}$$  (39) 
We apply Theorem 9, where
$$  
$F({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\frac{a}{4}{({y}_{1}+{z}_{1})}^{4}+\frac{\stackrel{~}{a}}{2}\left({y}_{1}^{2}{z}_{1}^{2}\right)+\frac{a}{2}{({y}_{1}+{z}_{1})}^{2}{\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
$G({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\frac{a}{4}{({y}_{2}+{z}_{2})}^{4}+\frac{\stackrel{~}{a}}{2}\left({y}_{2}^{2}{z}_{2}^{2}\right)+\frac{a}{2}{({y}_{2}+{z}_{2})}^{2}{\displaystyle \frac{1}{{y}_{1}^{2}+{z}_{1}^{2}+1}}.$ 
Note that both $F({x}_{1},{x}_{2})$ and $G({x}_{1},{x}_{2})$ (${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}$) are upperbounded by fourthdegree polynomials in ${x}_{1}$ and ${x}_{2}$, respectively, which ensures that (H1) is satisfied.
Verification of the condition (H2)’. Since $G({x}_{1},{x}_{2})=F({x}_{2},{x}_{1})$, similar reasoning as in the verification of (H2) from Example 1 leads to the conclusion that (H2)’ holds true.
Verification of the condition (H3)’. Given that the leading term in $G({x}_{1},{x}_{2})$ is a fourth degree polynomial in ${x}_{2}$, and that $G(\cdot ,{x}_{2})$ is bounded for each ${x}_{2}$, there exists a positive number $R$ such that
$$G({x}_{1},{x}_{2})\le 0,\text{for all}{x}_{2}\ge R.$$ 
Therefore, we can find another positive number $M$ such that for all ${x}_{1},{x}_{2}\in {\mathbb{R}}^{2}$, we have
$$G({x}_{1},{x}_{2})\le M\le \frac{\sigma}{2}{{x}_{2}}^{2}+M.$$ 
Verification of the condition (H4)’. First note that
${f}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{1}+{z}_{1})}^{3}+\stackrel{~}{a}{y}_{1}+a({y}_{1}+{z}_{1}){\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
${f}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{1}+{z}_{1})}^{3}\stackrel{~}{a}{z}_{1}+a({y}_{1}+{z}_{1}){\displaystyle \frac{1}{{y}_{2}^{2}+{z}_{2}^{2}+1}},$  
${g}_{1}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{2}+{z}_{2})}^{3}+\stackrel{~}{a}{y}_{2}+a({y}_{2}+{z}_{2}){\displaystyle \frac{1}{{y}_{1}^{2}+{z}_{1}^{2}+1}},$  
${g}_{2}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=a{({y}_{2}+{z}_{2})}^{3}\stackrel{~}{a}{z}_{2}+a({y}_{2}+{z}_{2}){\displaystyle \frac{1}{{y}_{1}^{2}+{z}_{1}^{2}+1}},$ 
which gives
$\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\stackrel{~}{a}{y}_{1}+\stackrel{~}{a}{z}_{1},$  
$\stackrel{~}{g}({y}_{1},{z}_{1},{y}_{2},{z}_{2})=\stackrel{~}{a}{y}_{2}+\stackrel{~}{a}{z}_{2}.$ 
The linearity of $\stackrel{~}{f}$ and $\stackrel{~}{g}$ yields
$\left(\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{f}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({y}_{1}{\overline{y}}_{1})$  
$\le \stackrel{~}{a}{{y}_{1}{\overline{y}}_{1}}^{2}+\stackrel{~}{a}\left{y}_{1}{\overline{y}}_{1}\right{z}_{1}{\overline{z}}_{1},$  
$\left(\stackrel{~}{f}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{f}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({z}_{1}{\overline{z}}_{1})$  
$\le \stackrel{~}{a}{{z}_{1}{\overline{z}}_{1}}^{2}+\stackrel{~}{a}\left{y}_{1}{\overline{y}}_{1}\right{z}_{1}{\overline{z}}_{1},$  
$\left(\stackrel{~}{g}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{g}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({y}_{2}{\overline{y}}_{2})$  
$\le \stackrel{~}{a}{{y}_{2}{\overline{y}}_{2}}^{2}+\stackrel{~}{a}{y}_{2}{\overline{y}}_{2}{z}_{2}{\overline{z}}_{2},$  
$\left(\stackrel{~}{g}({y}_{1},{z}_{1},{y}_{2},{z}_{2})\stackrel{~}{g}({\overline{y}}_{1},{\overline{z}}_{1},{\overline{y}}_{2},{\overline{z}}_{2})\right)({z}_{2}{\overline{z}}_{2})$  
$\le \stackrel{~}{a}{{z}_{2}{\overline{z}}_{2}}^{2}+\stackrel{~}{a}{y}_{2}{\overline{y}}_{2}{z}_{2}{\overline{z}}_{2}.$ 
Hence the monotonicity conditions ((H4)) hold with
${m}_{11}$  $=$  $\stackrel{~}{a},{m}_{12}=\stackrel{~}{a},{m}_{13}=0,{m}_{14}=0,$  
${m}_{21}$  $=$  $\stackrel{~}{a},{m}_{22}=\stackrel{~}{a},{m}_{23}=0,{m}_{24}=0,$  
${m}_{31}$  $=$  $0,{m}_{32}=0,{m}_{33}=\stackrel{~}{a},{m}_{34}=\stackrel{~}{a},$  
${m}_{41}$  $=$  $0,{m}_{42}=0,{m}_{43}=\stackrel{~}{a},{m}_{44}=\stackrel{~}{a}.$ 
Verification of the condition (H5). Simple computations yield
$$M=\left[\begin{array}{cc}1\beta \left(12\frac{\stackrel{~}{a}}{{\lambda}_{1}}\right)& 0\\ 0& 1\beta \left(12\frac{\stackrel{~}{a}}{{\lambda}_{1}}\right)\end{array}\right].$$ 
Since $12\frac{\stackrel{~}{a}}{{\lambda}_{1}}>0,$ we can choose $\beta >0$ in (31) sufficiently small that the matrix $M$ is convergent to zero.
Thus, all the hypothesis of Theorem 9 are satisfied and problem (39) has a solution $({v}_{1}^{\ast},{w}_{1}^{\ast},{v}_{2}^{\ast},{w}_{2}^{\ast}),$ where if ${u}_{1}^{\ast}:=$ $({v}_{1}^{\ast},{w}_{1}^{\ast})$ and ${u}_{2}^{\ast}:=({v}_{2}^{\ast},{w}_{2}^{\ast}),$ one has that ${u}_{1}^{\ast}$ is a mountain pass type critical point of the energy functional ${E}_{1}(\cdot ,{u}_{2}^{\ast}),$ and ${u}_{2}^{\ast}$ is a mountain pass type critical point of the energy functional ${E}_{2}({u}_{1}^{\ast},\cdot ).$
Acknowledgements
The authors would like to express their gratitude to the anonymous referees for their reviews and valuable remarks, which significantly improved the paper.
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