Linking Methods for Componentwise Variational Systems

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Abstract

The paper deals with the equilibrium solutions of two-equation 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 non-contradictory. 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, Babeş-Bolyai University, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Andrei Stan
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, 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/s00025-023-02026-x

About this paper

Journal

Results in Mathematics

Publisher Name

Springer

DOI

https://doi.org/10.1007/s00025-023-02026-x

Print ISSN

1422-6383

Online ISSN

1420-9012

google scholar link

[1] Ambrosetti, A., Rabinowitz, P.H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
[2] Beldinski, M., Galewski, M., Nash type equilibria for systems of non-potential equations. Appl. Math. Comput. 385, 125456 (2020)
[3] Benci, V., Rabinowitz, P.H., Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979)
[4] Boureanu, M.-M., Pucci, P., Radulescu, V.D., Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent. Complex Var. Elliptic Equ. 56, 755–767 (2011)
[5] Chabrowski, J., Introduction to the theory of critical points. The mountain pass theorem. Ekeland’s variational principle. In: Instructional Workshop on Analysis and Geometry, Part III, Canberra (1995)
[6] Costa, D.G., Magalhaes, C.A., Existence results for perturbations of the p -Laplacian. Nonlinear Anal. 24, 409–418 (1995)
[7] Costea, N., Csirik, M., Varga, C., Linking-type results in nonsmooth critical point theory and applications. Set-Valued Var. Anal. 25, 333–356 (2017)
[8] De Figueiredo, D.G., Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989)
[9] Filippucci, R., Pucci, P., Robert, F., On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 91(2), 156–177 (2009)
[10] Galewski, M., On the mountain pass solutions to boundary value problems on the Sierpinski gasket. Results Math. 74, 167 (2019)
[11] Grossinho, M.R., Tersian, S.A., An Introduction to Minimax Theorems and Their Applications to Differential Equations. Springer, Dordrecht (2001)
[12] Jebelean, P., Moro¸sanu, Gh., Mountain pass type solutions for discontinuous perturbations of the vector p-Laplacian. Nonlinear Funct. Anal. Appl. 10(4), 591–611 (2005)
[13] Kassay, G., R˘adulescu, V.D., Equilibrium Problems and Applications. Academic Press (2019)
[14] Le Dret, H., Nonlinear Elliptic Partial Differential Equations. Springer, Berlin (2018)
[15] Motreanu, D., Motreanu, V.V., Papageorgiou, N.S., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
[16] Mugnai, D., Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. NoDEA Nonlinear Differ. Equ. Appl. 11, 379–391 (2004)
[17] Precup, R., Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal. 75, 834–851 (2012)
[18] Precup, R., Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3(4), 197–207 (2014)
[19] Precup, R., A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems. J. Math. Anal. Appl. 463, 412–431 (2018)
[20] Precup, R., Componentwise localization of critical points for functionals defined on product spaces. Topol. Methods Nonlinear Anal. 58, 51–77 (2021)
[21] Precup, R., Stan, A., Stationary Kirchhoff equations and systems with reaction terms. AIMS Math. 7(8), 15258–15281 (2022)
[22] Pucci, P., Radulescu, V., The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 9(3), 543–584 (2010)
[23] Rabinowitz, P.H., Minimax methods in critical point theory with applications to nonlinear partial differential equations. Conf. Board of Math Sci. 65, Amer. Math. Soc. (1986)
[24] Schechter, M., Linking Methods in Critical Point Theory. Birkhauser, Boston (1999)
[25] Silva, E.A.B., Existence and multiplicity of solutions for semilinear elliptic systems. Nonlinear Differ. Equ. Appl. 1, 339–363 (1994)
[26] Stan, A., Nonlinear systems with a partial Nash type equilibrium. Studia Univ. Babes-Bolyai Math. 66, 397–408 (2021)
[27] Struwe, M., Variational Methods. Springer, Berlin (1990)

Paper (preprint) in HTML form

Linking methods for componentwise variational systems

Radu Precup r.precup@math.ubbcluj.ro Andrei Stan andrei.stan@ubbcluj.ro

Abstract

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 real-world 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

\left\{\begin{array}[]{l}E_{u_{1}}\left(u_{1},u_{2}\right)=0\\ E_{u_{2}}\left(u_{1},u_{2}\right)=0,\end{array}\right.

where $E_{u_{1}},E_{u_{2}}$ are the partial derivatives of $E\left(u_{1},u_{2}\right)$ in each of the two variables. A wide range of variational techniques that are well-established 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 $E\left(u_{1},u_{2}\right)$

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, $E_{1}\left(u_{1},u_{2}\right)$ and $E_{2}\left(u_{1},u_{2}\right)$ , and aim to find solutions to the system

\left\{\begin{array}[]{l}E_{11}\left(u_{1},u_{2}\right)=0\\ E_{22}\left(u_{1},u_{2}\right)=0,\end{array}\right.

(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 $\left(u_{1},u_{2}\right)$ in one of the following situations:

(a): The pair $\left(u_{1},u_{2}\right)$ is a Nash (min-min) equilibrium of the system, that is $u_{1}$ minimizes the functional $E_{1}\left(\cdot,u_{2}\right)$ and $u_{2}$ minimizes $E_{2}\left(u_{1},\cdot\right);$
(b): The pair $\left(u_{1},u_{2}\right)$ is a min-mountain pass equilibrium of the system, that is $u_{1}$ minimizes the functional $E_{1}\left(\cdot,u_{2}\right)$ and $u_{2}$ is a mountain pass type point of $E_{2}\left(u_{1},\cdot\right);$
(c): The pair $\left(u_{1},u_{2}\right)$ is a mountain pass-mountain pass equilibrium of the system, that is $u_{1}$ is a mountain pass type point of $E_{1}\left(\cdot,u_{2}\right)$ and $u_{2}$ is a mountain pass type point of $E_{2}\left(u_{1},\cdot\right).$

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 $u_{1}=\left(x,y\right)$ and $u_{2}=\left(z,w\right):$

(a)

	$\displaystyle E_{1}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}+y^{2}+z^{2}+w^{2}-xz,$
	$\displaystyle E_{2}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}+2y^{2}+z^{2}+w^{2}-yw.$

It is easy to see that $u_{1}=\left(0,0\right)$ and $u_{2}=\left(0,0\right)$ solves (1) and that $u_{1}$ minimizes $E_{1}\left(\cdot,u_{2}\right)=x^{2}+y^{2},$ while $u_{2}$ minimizes $E_{2}\left(u_{1},\cdot\right)=z^{2}+w^{2}.$

(b)

	$\displaystyle E_{1}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}+y^{2}+z^{2}+w^{2}-xz,$
	$\displaystyle E_{2}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}+2y^{2}+z^{2}-w^{2}-yw.$

Here again, $u_{1}=\left(0,0\right)$ and $u_{2}=\left(0,0\right)$ solves (1), and $u_{1}$ minimizes $E_{1}\left(\cdot,u_{2}\right)=x^{2}+y^{2},$ while $u_{2}$ is a mountain pass of $E_{2}\left(u_{1},\cdot\right)=z^{2}-w^{2}.$

(c)

	$\displaystyle E_{1}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}-y^{2}+z^{2}+w^{2}-xz,$
	$\displaystyle E_{2}\left(x,y,z,w\right)$	$\displaystyle=$	$\displaystyle x^{2}+2y^{2}+z^{2}-w^{2}-yw.$

In this case $u_{1}=\left(0,0\right)$ is a mountain pass type point of $E_{1}\left(\cdot,u_{2}\right)=x^{2}-y^{2}$ and $u_{2}=\left(0,0\right)$ is a mountain pass type point of $E_{2}\left(u_{1},\cdot\right)=z^{2}-w^{2}.$

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 min-max 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 $\emptyset\neq Q\subset D.$

Definition 1 ((20)).

We say that a nonempty set $A\subset D$ links a set $B\subset Q$ via $Q$ (in $D)$ if $\gamma\left(Q\right)\cap A\neq\emptyset$ for every $\gamma\in C\left(Q,D\right)$ with $\left.\gamma\right|_{B}=\$ id ${}_{B}.$

Note that, in virtue of the above definition, the total set $A=D$ links the empty set $B=\emptyset,$ 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 min-max procedure.

Assume that $A$ links $B$ in $D$ via $Q,$ let $\Gamma=\{\gamma\in C\left(Q,D\right):\ \left.\gamma\right|_{B}=\$ id ${}_{B}\},$ and $E:D\rightarrow\mathbb{R}$ be any functional. Denote

m:=\inf_{v\in D}E\left(v\right),\ \ \ a:=\inf_{v\in A}E\left(v\right),\ \ \ b:% =\sup_{v\in B}E\left(v\right),

and

c:=\inf_{\gamma\in\Gamma}\sup_{q\in Q}E\left(\gamma\left(q\right)\right).

Then it is easy to see that

m\leq a\leq c\ \ \ \text{and\ \ \ }b\leq c.

If $B=\emptyset$ and $A=D,$ then

m=a,\ \ b=-\infty\ \ \text{and \ }c=m.

The first equalities are obvious. To prove that $c=m,$ observe that in this case of trivial linking, $\Gamma$ is the whole space $C\left(Q,D\right)$ 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=\inf_{\gamma\in\Gamma}\sup_{q\in Q}E\left(\gamma\left(q\right)\right)\leq% \inf_{v\in D}\sup_{q\in Q}E\left(\gamma_{v}\left(q\right)\right)=\inf_{v\in D}% 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, $E_{1}\left(u_{1},u_{2}\right)$ and $E_{2}\left(u_{1},u_{2}\right),$ defined on a product space $X_{1}\times X_{2}.$ Correspondingly, we shall use one linking for the functionals $E_{1}\left(\cdot,u_{2}\right)$ with a fixed $u_{2}\in X_{2},$ and an other linking for the functionals $E_{2}\left(u_{1},\cdot\right)$ 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 $\Phi:X\rightarrow\mathbb{R}\cup\{+\infty\}$ be a lower semicontinuous and bounded below functional. Then, given any $\varepsilon>0$ , there exists $u_{\varepsilon}\in X$ such that

\Phi(u_{\varepsilon})\leq\inf_{X}\Phi+\varepsilon

and

\Phi(u_{\varepsilon})\leq\Phi(u)+\varepsilon d(u,u_{\varepsilon}),

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 $\left(X,\left|\cdot\right|_{X}\right)$ be a Banach space, $K$ a compact and $f\in C\left(K,X^{\ast}\right).$ Then, for each $\varepsilon>0$ , there exists a function $\varphi\in C\left(K,X\right)$ such that:

\left|\varphi\left(x\right)\right|_{X}\leq 1,\text{ and }\left(f\left(x\right)% ,\ \varphi\left(x\right)\right)>\left|f\left(x\right)\right|_{X}-\varepsilon,

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 $\varepsilon>0$ , there exists an $x_{\varepsilon}\in X$ that satisfies

|x_{\varepsilon}|_{X}\leq 1\quad\text{and}\quad(x^{*},x_{\varepsilon})>|x_{% \varepsilon}|\varepsilon.

Let $\varepsilon>0$ . According to the previous remark, for any $x_{0}\in K$ , there is $u_{0}\in X$ with $\left|u_{0}\right|_{X}\leq 1$ and $\left(f\left(x_{0}\right),u_{0}\right)>\left|f\left(x_{0}\right)\right|-\varepsilon.$ Define

U\left(x_{0}\right)=\left\{x\in K:\ \left(f\left(x\right),u_{0}\right)>\left|f% \left(x\right)\right|_{X}-\varepsilon\right\},

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 $K :$ $U\left(x_{1}\right),U\left(x_{2}\right),...,U\left(x_{n}\right).$ Let $u_{i}$ $(i=1,2,...,n)$ be the corresponding elements, i.e.,

U\left(x_{i}\right)=\left\{x\in K:\ \left(f\left(x\right),u_{i}\right)>\left|f% \left(x\right)\right|_{X}-\varepsilon\right\}.

Let $\rho_{i}\left(x\right)=\$ dist $\left(x,K\setminus U\left(x_{i}\right)\right)$ and $\zeta_{i}\left(x\right)=\rho_{i}\left(x\right)/\sum_{j=1}^{n}\rho_{j}\left(x% \right).$ Notice that $\zeta_{i}:K\rightarrow K$ is continuous, $\zeta_{i}\left(x\right)\neq 0$ 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

\varphi\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 $\left(x_{k,p}\right)_{k\geq 1},\ \left(y_{k,p}\right)_{k\geq 1}$ be two sequences of nonnegative real numbers depending on a parameter $p$ such that

\begin{bmatrix}x_{k,p}\\ y_{k,p}\end{bmatrix}\leq A\begin{bmatrix}0\\ y_{k-1,p}\end{bmatrix}+\left[\begin{array}[]{c}z_{k,p}\\ w_{k,p}\end{array}\right],

for all $k$ and $p,$ where $\left(z_{k,p}\right)_{k\geq 1},\left(w_{k,p}\right)_{k\geq 1}$ are sequences convergent to zero uniformly with respect to $p$ . If the matrix $A$ is convergent to zero and the sequence $\left(y_{k,p}\right)_{k\geq 1}$ is bounded uniformly with respect to $p$ , then $x_{k,p}\rightarrow 0$ and $y_{k,p}\rightarrow 0$ as $k\rightarrow\infty$ 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 $k\rightarrow\infty.$ The same conclusion holds if the spectral radius of the matrix is less than one, or if the inverse of $I-A$ (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

\text{tr}\left(A\right)<\min\left\{2,\ 1+\text{det}\left(A\right)\right\}.

Also, one can easily check that if $A=\left[a_{ij}\right]_{1\leq i,j\leq 2}$ is convergent to zero and

A^{\prime}:=\begin{bmatrix}a_{11}&0\\ a_{21}&a_{22}\end{bmatrix},\quad A^{\prime\prime}:=A-A^{\prime}=\begin{bmatrix% }0&a_{12}\\ 0&0\end{bmatrix},

(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 Nash-type 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 $\left(\cdot,\cdot\right)_{i}$ 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 $A_{i},Q_{i}\neq\emptyset$ and $B_{i}\subset Q_{i}.$ Denote

\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}}):=\max_{q\in Q_{i}}|\gamma_{i}(q)-% \overline{\gamma_{i}}(q)\ vert_{i},

for any $\gamma_{i},\overline{\gamma_{i}}\in\Gamma_{i}.$ Furthermore, for two functionals $E_{i}:H\rightarrow\mathbb{R}$ and each point $(u_{1},u_{2})\in H,$ we define:

\begin{array}[]{ll}m_{1}(u_{2}):=\inf_{X_{1}}E_{1}(\cdot,u_{2})\,;&m_{2}(u_{1}% ):=\inf_{X_{2}}E_{2}(u_{1},\cdot)\,;\\ a_{1}(u_{2}):=\inf_{A_{1}}E_{1}(\cdot,u_{2})\,;&a_{2}(u_{1}):=\inf_{A_{2}}E_{2% }(u_{1},\cdot)\,;\\ b_{1}(u_{2}):=\sup_{B_{1}}E_{1}(\cdot,u_{2})\,;&b_{2}(u_{1}):=\sup_{B_{2}}E_{2% }(u_{1},\cdot)\,;\\ \end{array}

	$\displaystyle c_{1}(u_{2}):=\inf_{\mu\in\Gamma_{1}}\max_{q\in Q_{1}}E_{1}(\mu(% q),u_{2})\,;$
	$\displaystyle c_{2}(u_{1}):=\inf_{\mu\in\Gamma_{2}}\max_{q\in Q_{2}}E_{2}(u_{1% },\mu(q)).$

As noted above, for each $i\in\{1,2\},$ one has

m_{i}\leq a_{i}\leq c_{i}\ \ \text{ and\ \ }b_{i}\leq 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 $i\in\left\{1,2\right\},$ let $A_{i}$ links $B_{i}$ via $Q_{i}$ in $H_{i}.$ If

b_{i}<a_{i},\ \ \ i=1,2,

then there exist two sequences $u_{1}^{k}\in H_{1}$ and $u_{2}^{k}\in H_{2}$ such that

0\leq E_{1}(u_{1}^{k},u_{2}^{k-1})-c_{1}\left(u_{2}^{k-1}\right)\rightarrow 0,% \ \ \ \ 0\leq E_{2}(u_{1}^{k},u_{2}^{k})-c_{2}\left(u_{1}^{k}\right)\rightarrow 0

(3)

and

E_{11}\left(u_{1}^{k},u_{2}^{k-1}\right)\rightarrow 0,\ \ \ \ E_{22}\left(u_{1% }^{k},u_{2}^{k}\right)\rightarrow 0,

(4)

as $k\rightarrow\infty.$

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 $\mathcal{E}_{1}\colon\Gamma_{1}\rightarrow\mathbb{R},$

\mathcal{E}_{1}(\mu)=\max_{Q_{1}}E_{1}(\mu(\cdot),u_{2}^{k-1})\ \ \left(\mu\in% \Gamma_{1}\right),

and observe that it is semi-continuous and bounded from below, since

\mathcal{E}_{1}(\gamma_{1})\geq a_{1}(u_{2}^{k-1})>b_{1}\left(u_{2}^{k-1}% \right)\geq-\infty

Thus, Lemma 1 guarantees the existence of a path $\gamma_{1}^{k}\in\Gamma_{1}$ such that

\mathcal{E}_{1}(\gamma_{1}^{k})\leq\inf_{\mu\in\Gamma_{1}}\mathcal{E}_{1}(\mu)% +\frac{1}{k}=c_{1}\left(u_{2}^{k-1}\right)+\frac{1}{k},

(5)

\mathcal{E}_{1}(\gamma_{1}^{k})-\mathcal{E}_{1}(\mu)\leq\frac{1}{k}d_{1}(% \gamma_{1}^{k},\mu),

(6)

for all $\mu\in\Gamma_{1}.$ If we consider

Q_{1}^{k}:=\left\{q_{1}\in Q_{1}\,:\ \,\mathcal{E}_{1}\left(\gamma_{1}^{k}(q_{% 1})\right)=E_{1}\left(\gamma_{1}^{k}(q_{1}),u_{2}^{k-1}\right)\right\},

one can see that $B_{1}\cap Q_{1}^{k}=\emptyset$ , since $b_{1}(u_{2}^{k-1})<a_{1}(u_{2}^{k-1}).$

Next we prove that there exists $q_{1}^{k}\in Q_{1}^{k}$ with $\left|E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right),v_{0}\right)\right|_{1}% <1/k.$ To this end we apply Lemma 2 to the function

f\left(q_{1}\right)=E_{11}\left(\gamma_{1}^{k}\left(\mu\right),u_{2}^{k-1}% \right),

from where we deduce the existence of a function $\varphi\in C\left(Q_{1},H_{1}\right)$ with $\left|\varphi\left(q_{1}\right)\right|_{1}\leq 1$ and

\left(E_{11}\left(\gamma_{1}^{k}\left(q_{1}\right),u_{2}^{k-1}\right),\ % \varphi\left(q_{1}\right)\right)_{1}>\left|E_{11}\left(\gamma_{1}^{k}\left(q_{% 1}\right),u_{2}^{k-1}\right)\right|_{1}-\frac{1}{k}\ \ \ \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)\varphi\left(q_{1}\right),

where $\zeta:Q_{1}\rightarrow\left[0,1\right]$ is continuous, $\zeta\left(q_{1}\right)=1$ on $Q_{1}^{k}$ and $\zeta=0$ on $B_{1}.$ We have $d_{1}\left(\gamma_{1}^{k},\eta\right)=\lambda\left|w\right|_{\infty}\leq\lambda$ and

\psi\left(\eta\right)=\max_{q_{1}\in Q_{1}}E_{1}\left(\eta\left(q_{1}\right),u% _{2}^{k-1}\right)=E_{1}\left(\eta\left(q_{1}^{\lambda}\right),u_{2}^{k-1}% \right),

for some $q_{1}^{\lambda}\in Q_{1}.$ Hence from (6), one has

E_{1}\left(\eta\left(q_{1}^{\lambda}\right),u_{2}^{k-1}\right)-\max_{q_{1}\in Q% _{1}}E_{1}\left(\gamma_{1}^{k}\left(q_{1}\right),u_{2}^{k-1}\right)+\frac{% \lambda}{k}\geq 0.

Since

	$\displaystyle E_{1}\left(\eta\left(q_{1}^{\lambda}\right),u_{2}^{k-1}\right)-E% _{1}\left(\gamma_{1}^{k}\left(q_{1}^{\lambda}\right),u_{2}^{k-1}\right)$
	$\displaystyle=-\lambda\left(E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{\lambda}% \right),u_{2}^{k-1}\right),w\left(q_{1}^{\lambda}\right)\right)_{1}+o\left(% \lambda\right)$

we deduce that

-\left(E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{\lambda}\right),u_{2}^{k-1}% \right),w\left(q_{1}^{\lambda}\right)\right)_{1}+\frac{1}{k}+\frac{1}{\lambda}% o\left(\lambda\right)\geq 0.

We may assume that $q_{1}^{\lambda}\rightarrow q_{1}^{k}\in Q_{1}^{k}$ as $\lambda\rightarrow 0.$ Then

-\left(E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right),u_{2}^{k-1}\right),w% \left(q_{1}^{k}\right)\right)_{1}+\frac{1}{k}\geq 0.

Thus, also using (7) and since $w\left(q_{1}^{k}\right)=\varphi\left(q_{1}^{k}\right),$ we have

\left|E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right),u_{2}^{k-1}\right)% \right|_{1}-\frac{1}{k}<\left(E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right)% ,u_{2}^{k-1}\right),w\left(q_{1}^{k}\right)\right)_{1}\leq\frac{1}{k},

whence

\left|E_{11}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right),u_{2}^{k-1}\right)% \right|_{1}<\frac{2}{k}.

We denote

u_{1}^{k}=\gamma_{1}^{k}\left(q_{1}^{k}\right).

Thus we have

E_{1}\left(u_{1}^{k},u_{2}^{k-1}\right)\leq c_{1}\left(u_{2}^{k-1}\right)+% \frac{1}{k},\ \ \ \left|E_{11}\left(u_{1}^{k},u_{2}^{k-1}\right)\right|_{1}<% \frac{2}{k}.

(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 $\mathcal{E}_{2}\colon\Gamma_{2}\rightarrow\mathbb{R},$

\mathcal{E}_{2}(\mu)=\max_{q_{2}\in Q_{2}}E_{2}(u_{1}^{k},\mu\left(q_{2}\right% ))\ \ \left(\mu\in\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 $\gamma_{2}^{k}\in\Gamma_{2}$ and

q_{2}^{k}\in Q_{2}^{k}=\left\{q_{2}\in Q_{2}\,:\ \,\mathcal{E}_{2}\left(\gamma% _{2}^{k}(q_{2})\right)=E_{2}\left(u_{1}^{k},\gamma_{2}^{k}(q_{2})\right)\right\},

having the properties

E_{2}\left(u_{1}^{k},u_{2}^{k}\right)\leq c_{2}\left(u_{1}^{k}\right)+\frac{1}% {k},\ \ \ \left|E_{22}\left(u_{1}^{k},u_{2}^{k}\right)\right|_{2}<\frac{2}{k}.

(9)

Clearly, (8) and (9) imply (3) and (4).

∎

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\,,\quad E_{22}(u^{\ast},v^{\ast})=0,

(10)

c_{1}\left(u_{2}^{k}\right)\rightarrow c_{1}\left(v^{\ast}\right),\ \ \ c_{2}% \left(u_{1}^{k}\right)\rightarrow c_{2}\left(u^{\ast}\right)

(11)

and

E_{1}(u^{\ast},v^{\ast})=c_{1}\left(v^{\ast}\right)\,,\quad 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: $c_{1}\left(u_{2}^{k-1}\right)\rightarrow E_{1}\left(u^{\ast},v^{\ast}\right).$ Indeed, one has

	$\displaystyle c_{1}\left(u_{2}^{k-1}\right)$	$\displaystyle=$	$\displaystyle\inf_{\mu\in\Gamma_{1}}\max_{q_{1}\in Q_{1}}E_{1}\left(\mu\left(q% _{1}\right),u_{2}^{k-1}\right)\leq\max_{q_{1}\in Q_{1}}E_{1}\left(\gamma_{1}^{% k}\left(q_{1}\right),u_{2}^{k-1}\right)$
		$\displaystyle=$	$\displaystyle E_{1}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right),u_{2}^{k-1}% \right)=E_{1}\left(u_{1}^{k},u_{2}^{k-1}\right)\leq c_{1}\left(u_{2}^{k-1}% \right)+\frac{1}{k}.$

Hence

	$\displaystyle E_{1}\left(u_{1}^{k},u_{2}^{k-1}\right)-E_{1}\left(u^{\ast},v^{% \ast}\right)-\frac{1}{k}$	$\displaystyle\leq$	$\displaystyle c_{1}\left(u_{2}^{k-1}\right)-E_{1}\left(u^{\ast},v^{\ast}\right)$
		$\displaystyle\leq$	$\displaystyle E_{1}\left(u_{1}^{k},u_{2}^{k-1}\right)-E_{1}\left(u^{\ast},v^{% \ast}\right),$

whence passing to the limit we deduce that $c_{1}\left(u_{2}^{k-1}\right)-E_{1}\left(u^{\ast},v^{\ast}\right)\rightarrow 0,$ as claimed.

Step 2: $E_{1}\left(u^{\ast},v^{\ast}\right)\leq c_{1}\left(v^{\ast}\right).$ Let $\mu\in\Gamma_{1}$ be any path. Then for each $k,$ there is $\overline{q}_{1}^{k}\in Q_{1}$ with

c_{1}\left(u_{2}^{k-1}\right)\leq\max_{q_{1}\in Q_{1}}E_{1}\left(\mu\left(q_{1% }\right),u_{2}^{k-1}\right)=E_{1}\left(\mu\left(\overline{q}_{1}^{k}\right),u_% {2}^{k-1}\right).

Since $Q_{1}$ is compact, passing to a subsequence we may assume that $\overline{q}_{1}^{k}\rightarrow q_{1}^{\mu}$ as $k\rightarrow\infty.$ Then taking the limit in the last inequality and using the conclusion from Step 1, we derive

E_{1}\left(u^{\ast},v^{\ast}\right)\leq E_{1}\left(\mu\left(\overline{q}_{1}^{% \mu}\right),v^{\ast}\right)\leq\max_{q_{1}\in Q_{1}}E_{1}\left(\mu\left(q_{1}% \right),v^{\ast}\right),

whence taking the infimum over $\mu\in\Gamma_{1}$ we obtain the desired inequality.

Step 3: $E_{1}\left(u^{\ast},v^{\ast}\right)\geq c_{1}\left(v^{\ast}\right).$ From the definition of $c_{1},$ one clearly has $c_{1}\left(v^{\ast}\right)\leq E_{1}\left(\gamma_{1}^{k}\left(q_{1}^{k}\right)% ,v^{\ast}\right)=E_{1}\left(u_{1}^{k},v^{\ast}\right)$ for all $k .$ Let $\varepsilon>0$ be arbitrarily chosen. Since $u_{2}^{k}\rightarrow v^{\ast},$ there exists $j_{k}$ such that $c_{1}\left(v^{\ast}\right)-\varepsilon\leq E_{1}\left(u_{1}^{k},v_{j}\right)$ for all $j\geq j_{k}.$ Thus, we can assume that $j_{k}>j_{k-1}$ and so that $j_{k}\rightarrow\infty$ as $k\rightarrow\infty.$ Then, from

c_{1}\left(v^{\ast}\right)-\varepsilon\leq E_{1}\left(u_{1}^{k},v_{j_{k}}% \right),

letting $k$ go to infinity, we deduce

c_{1}\left(v^{\ast}\right)-\varepsilon\leq E_{1}\left(u^{\ast},v^{\ast}\right).

Now since $\varepsilon$ is arbitrary, we must have $c_{1}\left(v^{\ast}\right)\leq E_{1}\left(u^{\ast},v^{\ast}\right),$ as claimed.

Finally, the two contrary inequalities in Steps 2 and 3 show that $c_{1}\left(v^{\ast}\right)=E_{1}\left(u^{\ast},v^{\ast}\right).$ ∎

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 $E_{2}\left(\cdot,v^{\ast}\right)$ and $v^{\ast}$ is a minimizer of the functional $E_{2}\left(u^{\ast},\cdot\right),$ that is the couple $\left(u^{\ast},v^{\ast}\right)$ 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 $E_{1}\left(\cdot,v^{\ast}\right),$ while $v^{\ast}$ is a minimizer of the functional $E_{2}\left(u^{\ast},\cdot\right).$

(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 $E_{2}\left(\cdot,v^{\ast}\right)$ and $v^{\ast}$ is a mountain pass type point of the functional $E_{2}\left(u^{\ast},\cdot\right).$

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 $L=\left(L_{1},L_{2}\right):H\rightarrow H,\ L_{i}:H\rightarrow H_{i}\ \left(i=% 1,2\right)$ be a continuous operator and let $N=\left(N_{1},N_{2}\right):H\rightarrow H,$ $N_{i}:H\rightarrow H_{i}$ $\left(i=1,2\right),$ be defined by

N\left(u\right)=u-L\left(E_{11}\left(u\right),E_{22}\left(u\right)\right).

(13)

Assume that the following conditions are satisfied:

(i) there are nonnegative constants $a_{ij}$ $\left(i,j=1,2\right)$ such that

	$\displaystyle\left(N_{1}(u_{1},u_{2})-N_{1}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{1}-\overline{u}_{1}\right)_{1}$		(14)
	$\displaystyle\leq a_{11}\,\|u_{1}-\overline{u}_{1}\|_{1}^{2}+a_{12}\left\|u_{1}-% \overline{u}_{1}\right\|_{1}\,\left\|u_{2}-\overline{u}_{2}\right\|_{2},$
	$\displaystyle\left(N_{2}(u_{1},u_{2})-N_{2}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{2}-\overline{u}_{2}\right)_{2}$		(15)
	$\displaystyle\leq a_{22}\,\|u_{2}-\overline{u}_{2}\|_{2}^{2}+a_{21}\left\|u_{1}-% \overline{u}_{1}\right\|_{1}\,\left\|u_{2}-\overline{u}_{2}\right\|_{2},$

for all $u_{1},\overline{u}_{1}\in H_{1}$ and $u_{2},\overline{u}_{2}\in H_{2};$

(ii) the matrix $A=[a_{i,j}]_{1\leq i,j\leq 2}$ 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 $E_{11}\left(u_{1}^{k},u_{2}^{k-1}\right),\ E_{22}\left(u_{1}^{k},u_{2}^{k}\right)$ are convergent to zero, and the operators $L_{1},L_{2}$ are continuous, one has that the sequences

	$\displaystyle\alpha_{k}:=L_{1}(E_{11}\left(u_{1}^{k},u_{2}^{k-1}\right),\ E_{2% 2}\left(u_{1}^{k},u_{2}^{k}\right),$
	$\displaystyle\beta_{k}:=L_{1}(E_{11}\left(u_{1}^{k},u_{2}^{k-1}\right),\ E_{22% }\left(u_{1}^{k},u_{2}^{k}\right)$

are also convergent to zero. In terms of $\alpha_{k}$ and $\beta_{k},$ formula (13) gives

u_{1}^{k}=\alpha_{k}+N_{1}\left(u_{1}^{k},u_{2}^{k-1}\right),\ \ \ u_{2}^{k}=% \beta_{k}+N_{2}\left(u_{1}^{k},u_{2}^{k}\right).

Then, using the monotony conditions (14), we deduce

$\displaystyle\left\|u_{1}^{k}-u_{1}^{k+p}\right\|_{1}^{2}$	$\displaystyle=\left(u_{1}^{k}-u_{1}^{k+p}\,,\,\alpha_{k}-\alpha_{k+p}\right)_{1}$	(16)
	$\displaystyle\quad+\left(u_{1}^{k}-u_{1}^{k+p}\,,\,N_{1}(u_{1}^{k},u_{2}^{k-1}% )-N_{1}(u_{1}^{k+p},u_{2}^{k+p-1})\right)_{1}$
	$\displaystyle\leq a_{11}\left\|u_{1}^{k}-u_{1}^{k+p}\right\|_{1}^{2}+a_{12}\left% \|u_{1}^{k}-u_{1}^{k+p}\right\|_{1}\,\left\|u_{2}^{k-1}-u_{2}^{k+p-1}\right\|_{2}$
	$\displaystyle\quad+\left\|\alpha_{k}-\alpha_{k+p}\right\|_{1}\left\|u_{1}^{k}-u_{% 1}^{k+p}\right\|_{1}.$

Similarly,

	$\displaystyle\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}^{2}$	$\displaystyle\leq a_{22}\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}^{2}+a_{21}\left% \|u_{1}^{k}-u_{1}^{k+p}\right\|_{1}\,\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}$		(17)
		$\displaystyle\quad+\left\|\beta_{k}-\beta_{k+p}\right\|_{2}\left\|u_{2}^{k}-u_{2}% ^{k+p}\right\|_{2}.$

Under the notations

	$\displaystyle 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},\ \ \$
	$\displaystyle z_{k,p}:=\left\|\alpha_{k}-\alpha_{k+p}\right\|_{1},\ \ \ w_{k,p}:% =\left\|\beta_{k}-\beta_{k+p}\right\|_{2},$

inequalities (16)-(17) can be put under the matrix form

\begin{bmatrix}x_{k,p}\\ y_{k,p}\end{bmatrix}\leq A^{\prime}\begin{bmatrix}x_{k,p}\\ y_{k,p}\end{bmatrix}+A^{\prime\prime}\begin{bmatrix}0\\ y_{k-1,p}\end{bmatrix}+\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

\begin{bmatrix}x_{k,p}\\ y_{k,p}\end{bmatrix}\leq\widetilde{A}\begin{bmatrix}0\\ y_{k-1,p}\end{bmatrix}+\left(I-A^{\prime}\right)^{-1}\left[\begin{array}[]{c}z% _{k,p}\\ w_{k,p}\end{array}\right],

where the matrix $\widetilde{A}:=(I-A^{\prime})^{-1}A^{\prime\prime}$ is convergent to zero. Thus, Lemma 3 provides assurance that the sequences $\left(x_{k,p}\right)_{k\geq 1}$ and $\left(y_{k,p}\right)_{k\geq 1}$ 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 $\left(u_{2}^{k_{j}}\right)_{j\geq 1}$ is convergent, then it is not sure that the sequence $\left(u_{2}^{k_{j}-1}\right)_{j\geq 1}$ 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 Nash-type 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}\left(\cdot,w\right)\ \ \text{is bounded on }H_{1};\text{ }

(19)

E_{2}\left(u,\cdot\right)\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}\left(\cdot,w\right)\text{\ \ is bounded on }H_{1};

(21)

-E_{2}\left(u,\cdot\right)\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}\left(u_{1}^{k},u_{2}^{k}\right)\leq m_{2}\left(u_{1}^{k}\right)+\frac{1}% {k}\leq E_{2}\left(u_{1}^{k},w\right)+1\leq 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 $C\leq E_{2}\left(u_{1}^{k},w\right)$ for all $k .$ Since $w\in B_{2},$ one has $\gamma_{2}^{k}\left(w\right)=w.$ Then

	$\displaystyle C$	$\displaystyle\leq$	$\displaystyle E_{2}\left(u_{1}^{k},w\right)=E_{2}\left(u_{1}^{k},\gamma_{2}^{k% }\left(w\right)\right)\leq\max_{q_{2}\in Q_{2}}E_{2}\left(u_{1}^{k},\gamma_{2}% ^{k}\left(q_{2}\right)\right)$
		$\displaystyle=$	$\displaystyle E_{2}\left(u_{1}^{k},\gamma_{2}^{k}\left(q_{2}^{k}\right)\right)% =E_{2}\left(u_{1}^{k},u_{2}^{k}\right),$

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 Ambrosetti-Rabinowitz 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

(1⁰): $E_{1}=E_{2}=E;\ \$ or
(2⁰): $E_{1}=E$ and $E_{2}=-E.$

The results for case (2⁰) will be in some sense dual to those for case (1⁰). Thus, one can produce critical points $\left(u_{1}^{\ast},u_{2}^{\ast}\right)$ of $E,$ with one of the properties:

$\displaystyle E\left(u_{1}^{\ast},u_{2}^{\ast}\right)$	$\displaystyle=$	$\displaystyle\min E\left(\cdot,u_{2}^{\ast}\right)=\max E\left(u_{1}^{\ast},% \cdot\right);$
$\displaystyle E\left(u_{1}^{\ast},u_{2}^{\ast}\right)$	$\displaystyle=$	$\displaystyle\min E\left(\cdot,u_{2}^{\ast}\right)=\sup_{\mu\in\Gamma_{2}}\min% _{q\in Q_{2}}E\left(u_{1}^{\ast},\mu\left(q\right)\right);$
$\displaystyle E\left(u_{1}^{\ast},u_{2}^{\ast}\right)$	$\displaystyle=$	$\displaystyle\inf_{\mu\in\Gamma_{1}}\max_{q\in Q_{1}}E\left(\mu\left(q\right),% u_{2}^{\ast}\right)=\max E\left(u_{1}^{\ast},\cdot\right);$
$\displaystyle E\left(u_{1}^{\ast},u_{2}^{\ast}\right)$	$\displaystyle=$	$\displaystyle\inf_{\mu\in\Gamma_{1}}\max_{q\in Q_{1}}E\left(\mu\left(q\right),% u_{2}^{\ast}\right)=\sup_{\mu\in\Gamma_{2}}\min_{q\in Q_{2}}E\left(u_{1}^{\ast% },\mu\left(q\right)\right).$

3 Application

We apply the results from the previous section to the Dirichlet problem

\begin{cases}-\Delta v_{1}=\nabla_{v_{1}}F(v_{1},w_{1},v_{2},w_{2})\\ -\Delta w_{1}=\nabla_{w_{1}}F(v_{1},w_{1},v_{2},w_{2})\\ -\Delta v_{2}=\nabla_{v_{2}}G(v_{1},w_{1},v_{2},w_{2})\quad\\ -\Delta w_{2}=\nabla_{w_{2}}G(v_{1},w_{1},v_{2},w_{2})\ \ \ \text{on }\Omega\\ v_{1}|_{\partial\Omega}=w_{1}|_{\partial\Omega}=v_{2}|_{\partial\Omega}=w_{2}|% _{\partial\Omega}=0,\end{cases}

(23)

where $\Omega$ is a bounded open set from $\mathbb{R}^{n}$ $(n\geq 3)$ . These kinds of problems are widely recognized in the literature and they model real-world processes, such as stationary diffusion or wave propagation.

Throughout the section, the symbol $\left|\cdot\right|$ is used to denote the Euclidean norm in $\mathbb{R}^{2}$ . We assume the following behavior of potentials $F$ and $G :$

(H1)

$F,G\colon\mathbb{R}^{4}\rightarrow\mathbb{R}$ are of $C^{1}$ class and satisfy

$F(0,x_{2})=0\ \ \ \text{and\ \ \ }G(x_{1},0)=0,$

for all $x_{1},x_{2}\in\mathbb{R}^{2},$ and the growth conditions

$\displaystyle\left|F(x_{1},x_{2})\right|$ $\displaystyle\leq$ $\displaystyle C_{F}\left(\left|x_{1}\right|^{p}+1\right),$ (24)

$\displaystyle\left|G(x_{1},x_{2})\right|$ $\displaystyle\leq$ $\displaystyle C_{G}\left(\left|x_{2}\right|^{p}+1\right)\,,$

for all $x_{1},x_{2}\in\mathbb{R}^{2}$ and some positive constants $C_{F},C_{G},$ where $2\leq p\leq 2^{\ast}=\tfrac{2n}{n-2}.$

Here we take $H_{1}=H_{2}:=$ $\left(H_{0}^{1}(\Omega)\right)^{2}=H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)$ 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 $E_{1},E_{2}\colon\left(H_{0}^{1}(\Omega)\right)^{2}\times\left(H_{0}^{1}(% \Omega)\right)^{2}\rightarrow\mathbb{R},$

	$\displaystyle E_{1}(u_{1},u_{2})=\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1% }}^{2}-\int_{\Omega}F(u_{1},u_{2}),$
	$\displaystyle E_{2}(u_{1},u_{2})=\tfrac{1}{2}\|u_{2}\|_{H_{0}^{1}\times H_{0}^{1% }}^{2}-\int_{\Omega}G(u_{1},u_{2}),$

where $u_{1}=\left(v_{1},w_{1}\right),\ u_{2}=\left(v_{2},w_{2}\right)\in\left(H_{0}^% {1}(\Omega)\right)^{2}$ .

We are interested in a weak solution $\left(u_{1}^{\ast},u_{2}^{\ast}\right)$ of (23) such that $\left(u_{1}^{\ast},u_{2}^{\ast}\right)$ is a mountain pass-min 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

	$\displaystyle f_{1}(y_{1},z_{1},y_{2},z_{2})=\nabla_{y_{1}}F(y_{1},z_{1},y_{2}% ,z_{2}),$
	$\displaystyle f_{2}(y_{1},z_{1},y_{2},z_{2})=\nabla_{z_{1}}F(y_{1},z_{1},y_{2}% ,z_{2}),$
	$\displaystyle g_{1}(y_{1},z_{1},y_{2},z_{2})=\nabla_{y_{2}}G(y_{1},z_{1},y_{2}% ,z_{2}),$
	$\displaystyle g_{2}(y_{1},z_{1},y_{2},z_{2})=\nabla_{z_{2}}G(y_{1},z_{1},y_{2}% ,z_{2}),$

the identification of $H^{-1}(\Omega)$ with $H_{0}^{1}(\Omega)$ via $-\Delta$ yields to the representation

	$\displaystyle E_{11}(u_{1},u_{2})$	$\displaystyle=$	$\displaystyle u_{1}-\left((-\Delta)^{-1}f_{1}(u_{1},u_{2})\,,\,(-\Delta)^{-1}f% _{2}(u_{1},u_{2})\right),$
	$\displaystyle E_{22}(u_{1},u_{2})$	$\displaystyle=$	$\displaystyle u_{2}-\left((-\Delta)^{-1}g_{1}(u_{1},u_{2})\,,\,(-\Delta)^{-1}g% _{2}(u_{1},u_{2})\right).$

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 $\left(L^{2^{\ast}}(\Omega)\right)^{4}$ to $\left(L^{\left(2^{\ast}\right)^{\prime}}(\Omega)\right)^{2}$ , continuous and bounded (map bounded sets into bounded sets). Consequently, the operators

	$\displaystyle N_{1}(u_{1},u_{2})=\left((-\Delta)^{-1}f_{1}(u_{1},u_{2})),(-% \Delta)^{-1}f_{2}(u_{1},u_{2}))\right)$
	$\displaystyle N_{2}(u_{1},u_{2})=\left((-\Delta)^{-1}g_{1}(u_{1},u_{2})),(-% \Delta)^{-1}g_{2}(u_{1},u_{2}))\right)$

are well-defined and continuous from $\left(H_{0}^{1}(\Omega)\right)^{4}$ to $\left(H_{0}^{1}(\Omega)\right)^{2}$ .

Let $\lambda_{1}$ be the first eigenvalue of the Dirichlet problem $-\Delta u=\lambda v$ in $\Omega,\,v=0$ on $\partial\Omega$ (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

$\limsup_{\left|x_{1}\right|\rightarrow 0}\frac{F(x_{1},x_{2})}{\left|x_{1}% \right|^{2}}<\tfrac{\lambda_{1}}{2}<\liminf_{|y_{1}|\rightarrow\infty}\frac{F% \left((y_{1},0),x_{2}\right)}{y_{1}{}^{2}},$

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 $0<\mu<\lambda_{1}<\tau,$ and $C_{\mu},C_{\tau}>0$ such that

\tfrac{\tau}{2}y_{1}{}^{2}-C_{\tau}\leq F\left((y_{1},0),x_{2}\right)\ \text{ % and \ }F(x_{1},x_{2})\leq\tfrac{\mu}{2}\left|x_{1}\right|^{2}+C_{\mu}\left|x_{% 1}\right|^{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

$\displaystyle E_{1}\left((\alpha\phi_{1},0),u_{2}\right)$	$\displaystyle=\tfrac{1}{2}\left\|(\alpha\phi_{1},0)\right\|_{H_{0}^{1}\times H_{% 0}^{1}}^{2}-\int_{\Omega}F\left((\alpha\phi_{1},0),u_{2}\right)$	(26)
	$\displaystyle\leq\tfrac{1}{2}\alpha^{2}\|\phi_{1}\|_{H_{0}^{1}}^{2}-\tfrac{1}{2}% \tau\alpha^{2}\|\phi_{1}\|_{L^{2}}^{2}+C_{\tau}\,\text{meas}(\Omega)$
	$\displaystyle=\tfrac{1}{2}\left(1-\tfrac{\tau}{\lambda_{1}}\right)\alpha^{2}+C% _{\tau}\,\text{meas}(\Omega)\rightarrow-\infty,\ \text{ as }\alpha\rightarrow\infty,$

uniformly with respect to $u_{2}\in\left(H_{0}^{1}\left(\Omega\right)\right)^{2}$ , whilst the second one implies

	$\displaystyle E_{1}(u_{1},u_{2})$	$\displaystyle=\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\int_{\Omega% }F(u_{1},u_{2})$
		$\displaystyle\geq\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\tfrac{% \mu}{2}\int_{\Omega}\|u_{1}(x)\|^{2}-C_{\mu}\int_{\Omega}\|u_{1}(x)\|^{p}dx$
		$\displaystyle\geq\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\tfrac{% \mu}{2\lambda_{1}}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-C_{\mu}^{\prime}\|u_{% 1}\|_{H_{0}^{1}\times H_{0}^{1}}^{p}$
		$\displaystyle=\left(\tfrac{1}{2}-\tfrac{\mu}{2\lambda_{1}}\right)\|u_{1}\|_{H_{0% }^{1}\times H_{0}^{1}}^{2}-C_{\theta}^{\prime}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{% 1}}^{p}.$

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})\geq c>0\ \ \ \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

E_{1}\left((\alpha_{0}\phi_{1},0),u_{2}\right)<0\ \ \ \text{for all\ \ }u_{2}% \in\left(H_{0}^{1}(\Omega)\right)^{2}.

(28)

In addition, one has

E_{1}((0,0),u_{2})=0.

(29)

Now, if we consider in $\left(H_{0}^{1}(\Omega)\right)^{2}$ the sets

	$\displaystyle A_{1}=\left\{u_{1}\in\left(H_{0}^{1}(\Omega)\right)^{2}:\,\|u_{1}% \|_{H_{0}^{1}\times H_{0}^{1}}=r_{0}^{\prime}\right\},$
	$\displaystyle Q_{1}=\left\{s\left(\phi_{1},0\right)\in\left(H_{0}^{1}(\Omega)% \right)^{2}:\,0\leq s\leq\alpha_{0}\right\},$
	$\displaystyle B_{1}=\left\{\left(\left(0,0\right),\ \left(s_{0}\phi_{1},0% \right)\right)\right\},$

then, from (27), (28), (29), we see that $A_{1}$ links $B_{1}$ via $Q_{1}$ , and moreover

\inf_{A_{1}}E_{1}(\cdot,u_{2})\geq c>\sup_{B_{1}}E_{1}(\cdot,u_{2}),

for all $u_{2}\in\left(H_{0}^{1}(\Omega)\right)^{2}$ , i.e., $b_{1}<a_{1}.$

Also take

A_{2}=\left(H_{0}^{1}(\Omega)\right)^{2},\ \ \ B_{2}=\emptyset\ \ \ \text{and% \ \ }Q_{2}=\{(0,0)\},

which corresponds to the trivial linking. Furthermore, in order to have $\ b_{2}<a_{2},$ equivalently $-\infty<m_{2},$ 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 $\ 0\leq\sigma<\lambda_{1}$ with

$G\left(x_{1},x_{2}\right)\leq\frac{\sigma}{2}\left|x_{2}\right|^{2}+C,$ (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 $L=\left(L_{1},L_{2}\right),$ here linear, with $\ \ L_{1},L_{2}\colon\left(H_{0}^{1}(\Omega)\right)^{2}\rightarrow\left(H_{0}^% {1}(\Omega)\right)^{2}$ defined as

L_{1}(v_{1},w_{1})=L_{1}\left(u_{1}\right)=\beta(v_{1}-w_{1},v_{1}-w_{1}),% \quad\ L_{2}(v_{2},w_{2})=L_{2}\left(u_{2}\right)=u_{2},

(31)

for $u_{1}=\left(v_{1},w_{1}\right),\ u_{2}=(v_{2},w_{2})\in\left(H_{0}^{1}(\Omega)% \right)^{2}$ and some $\beta>0.$ Thus, correspondingly, one has

	$\displaystyle N_{1}(u_{1},u_{2})$
	$\displaystyle=u_{1}-L_{1}\,\left(E_{11}(u_{1},u_{2})\right)$
	$\displaystyle=u_{1}-L_{1}u_{1}+L_{1}\,\left((-\Delta)^{-1}f_{1}(u_{1},u_{2})),% (-\Delta)^{-1}f_{2}(u_{1},u_{2}))\right)$
	$\displaystyle=\left((1-\beta)v_{1}+\beta w_{1},(1-\beta)w_{1}-\beta v_{1}\right)$
	$\displaystyle\quad+\beta\left((-\Delta)^{-1}\left(f_{1}(u_{1},u_{2})-f_{2}(u_{% 1},u_{2})\right)\,,\,(-\Delta)^{-1}\left(f_{1}(u_{1},u_{2})-f_{2}(u_{1},u_{2})% \right)\right)$

and

	$\displaystyle N_{2}(u_{1},u_{2})$	$\displaystyle=u_{2}-L_{2}\,\left(E_{22}(u_{1},u_{2})\right)$
		$\displaystyle=u_{2}-L_{2}\,u_{2}+L_{2}\,\left((-\Delta)^{-1}g_{1}(u_{1},u_{2})% ),(-\Delta)^{-1}g_{2}(u_{1},u_{2}))\right)$
		$\displaystyle=\left((-\Delta)^{-1}g_{1}(u_{1},u_{2})\,,\,(-\Delta)^{-1}g_{2}(u% _{1},u_{2})\right)\quad\quad\quad\quad\,\,\,\,\;\;$

Next we introduce some monotonicity conditions on the functions $\widetilde{f}:=f_{1}-f_{2},$ $g_{1}$ 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

			$\displaystyle\left(\widetilde{f}(x_{1},x_{2})-\widetilde{f}(\overline{x}_{1},% \overline{x}_{2})\right)\left(y_{1}-\overline{y}_{1}\right)$
		$\displaystyle\leq$	$\displaystyle\|y_{1}-\overline{y}_{1}\|\left(m_{11}\|y_{1}-\overline{y}_{1}\|+m_{1% 2}\|z_{1}-\overline{z}_{1}\|+m_{13}\|y_{2}-\overline{y}_{2}\|+m_{14}\|z_{2}-% \overline{z}_{2}\|\right),$
			$\displaystyle\left(\widetilde{f}(x_{1},x_{2})-\widetilde{f}(\overline{x}_{1},% \overline{x}_{2})\right)\left(z_{1}-\overline{z}_{1}\right)$
		$\displaystyle\leq$	$\displaystyle\|z_{1}-\overline{z}_{1}\|\left(m_{21}\|y_{1}-\overline{y}_{1}\|+m_{2% 2}\|z_{1}-\overline{z}_{1}\|+m_{23}\|y_{2}-\overline{y}_{2}\|+m_{24}\|z_{2}-% \overline{z}_{2}\|\right),$
			$\displaystyle\left(g_{1}(x_{1},x_{2})-g_{1}(\overline{x}_{1},\overline{x}_{2})% \right)\left(y_{2}-\overline{y}_{2}\right)$
		$\displaystyle\leq$	$\displaystyle\|y_{2}-\overline{y}_{2}\|(m_{31}\|y_{1}-\overline{y}_{1}\|+m_{32}\|z_% {1}-\overline{z}_{1}\|+m_{33}\|y_{2}-\overline{y}_{2}\|+m_{34}\|z_{2}-\overline{z}% _{2}\|),$
			$\displaystyle\left(g_{2}(x_{1},x_{2})-g_{2}(\overline{x}_{1},\overline{x}_{2})% \right)\left(z_{2}-\overline{z}_{2}\right)$
		$\displaystyle\leq$	$\displaystyle\|z_{2}-\overline{z}_{2}\|\left(m_{41}\|y_{1}-\overline{y}_{1}\|+m_{4% 2}\|z_{1}-\overline{z}_{1}\|+m_{43}\|y_{2}-\overline{y}_{2}\|+m_{44}\|z_{2}-% \overline{z}_{2}\|\right),$

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:

$\displaystyle a_{11}$	$\displaystyle=1-\beta+\tfrac{\beta}{\lambda_{1}}\max\{m_{11},\ m_{22}\}+\tfrac% {\beta}{2\lambda_{1}}\left(m_{12}+m_{21}\right),$	(33)
$\displaystyle a_{12}$	$\displaystyle=\tfrac{\beta}{\lambda_{1}}\max\left\{\sqrt{m_{13}^{2}+m_{23}^{2}% },\ \sqrt{m_{14}^{2}+m_{24}^{2}}\right\},$	(34)
$\displaystyle a_{21}$	$\displaystyle=\frac{1}{\lambda_{1}}\max\left\{\sqrt{m_{31}^{2}+m_{32}^{2}},\ % \sqrt{m_{41}^{2}+m_{42}^{2}}\right\},$	(35)
$\displaystyle a_{22}$	$\displaystyle=\frac{m_{34}+m_{43}}{2\lambda_{1}}+\max\left\{m_{33},\ m_{44}% \right\}.$	(36)

Indeed, for any $u_{1}=\left(v_{1},w_{1}\right),u_{2},\overline{u}_{1},\overline{u}_{2}\in\left% (H_{0}^{1}(\Omega)\right)^{2},$ we have

	$\displaystyle\left(N_{1}(u_{1},u_{2})-N_{1}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{1}-\overline{u}_{1}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle=\left(1-\beta\right)\left\|u_{1}-\overline{u}_{1}\right\|_{H_{0}^{% 1}\times H_{0}^{1}}^{2}+\beta\left(\widetilde{f}(u_{1},u_{2})-\widetilde{f}(% \overline{u}_{1},\overline{u}_{2})\,,\,v_{1}-\overline{v}_{1}\right)_{L^{2}}$
	$\displaystyle\quad+\beta\left(\widetilde{f}(u_{1},u_{2})-\widetilde{f}(% \overline{u}_{1},\overline{u}_{2})\,,\,w_{1}-\overline{w}_{1}\right)_{L^{2}}.$

Using (H4) and the well known inequality $|v|_{L^{2}}|w|_{L^{2}}\leq\tfrac{1}{2}\left(|v|_{L^{2}}^{2}+|w|_{L^{2}}^{2}% \right),$ we obtain

	$\displaystyle\left(N_{1}(u_{1},u_{2})-N_{1}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{1}-\overline{u}_{1}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq\left(1-\beta\right)\left(\|v_{1}-\overline{u}_{1}\|_{H_{0}^{1}% }^{2}+\|w_{1}-\overline{w}_{1}\|_{H_{0}^{1}}^{2}\right)$
	$\displaystyle\quad+\beta m_{11}\|v_{1}-\overline{v}_{1}\|_{L^{2}}^{2}+\beta m_{1% 2}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{22}\|w_{1}-\overline{w}_{1}\|_{L^{2}}^{2}+\beta m_{2% 1}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{13}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}% \left\|v_{2}-\overline{v}_{2}\right\|_{L^{2}}+m_{14}\left\|v_{1}-\overline{v}_{1}% \right\|_{L^{2}}\left\|w_{2}-\overline{w}_{2}\right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{23}\left\|w_{1}-\overline{w}_{1}\right\|_{L^{2}}% \left\|v_{2}-\overline{v}_{2}\right\|_{L^{2}}+m_{24}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}\left\|w_{2}-\overline{w}_{2}\right\|_{L^{2}}.$

As both $|v_{1}-\overline{v}_{1}|_{L^{2}}$ and $\left|w_{1}-\overline{w}_{1}\right|_{L^{2}}$ are less or equal to $|u_{1}\mathbf{-}\overline{u}_{1}|_{L^{2}\times L^{2}},$ from Poincaré’s inequality $\ |v|_{L^{2}}^{2}\leq\tfrac{1}{\lambda_{1}}|v|_{H_{0}^{1}}^{2},$ we infer that

	$\displaystyle\left(N_{1}(u_{1},u_{2})-N_{1}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{1}-\overline{u}_{1}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq a_{11}\left\|u_{1}\mathbf{-}\overline{u}_{1}\right\|_{H_{0}^{1% }\times H_{0}^{1}}^{2}+a_{12}\|u_{1}\mathbf{-}\overline{u}_{1}\|_{H_{0}^{1}% \times H_{0}^{1}}\|u_{2}-\overline{u}_{2}\|_{H_{0}^{1}\times H_{0}^{1}}.$

Similarly, we have

	$\displaystyle\left(N_{2}(u_{1},u_{2})-N_{2}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{2}-\overline{u}_{2}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq m_{33}\|v_{2}-\overline{v}_{2}\|_{H_{0}^{1}}^{2}+m_{44}\|w_{2}-% \overline{w}_{2}\|_{H_{0}^{1}}^{2}$
	$\displaystyle\quad+\left(m_{34}+m_{43}\right)\|v_{2}-\overline{v}_{2}\|_{L^{2}}\|% w_{2}-\overline{w}_{2}\|_{L^{2}}$
	$\displaystyle\quad+\|v_{1}-\overline{v}_{1}\|_{L^{2}}\left(m_{31}\|v_{1}-% \overline{v}_{1}\|_{L^{2}}+m_{32}\|w_{1}-\overline{w}_{2}\|_{L^{2}}\right)$
	$\displaystyle\quad+\|v_{2}-\overline{v}_{2}\|_{L^{2}}\left(m_{41}\|v_{1}-% \overline{v}_{1}\|_{L^{2}}+m_{42}\|w_{1}-\overline{w}_{2}\|_{L^{2}}\right),$

which after further computation gives

	$\displaystyle\left(N_{2}(u_{1},u_{2})-N_{2}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{2}-\overline{u}_{2}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq a_{22}\|u_{2}-\overline{u}_{2}\|_{H_{0}^{1}\times H_{0}^{1}}^{% 2}+a_{21}\|u_{1}-\overline{u}_{1}\|_{H_{0}^{1}\times H_{0}^{1}}\|u_{2}-\overline{% u}_{2}\|_{H_{0}^{1}\times H_{0}^{1}}.$

Now it is clear that the first two conditions outlined in Theorem 6 are satisfied provided that

(H5)

The matrix $M:=[a_{ij}]_{1\leq i,j\leq 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 $u_{1}\in(H_{0}^{1}(\Omega))^{2}.$ Next, the growth condition (30) on $G$ gives

	$\displaystyle E_{2}\left(u_{1},u_{2}\right)$	$\displaystyle=\tfrac{1}{2}\|u_{2}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\int_{\Omega% }G(u_{1},u_{2})$
		$\displaystyle\geq\tfrac{1}{2}\|u_{2}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\frac{% \sigma}{2}\left\|u_{2}\right\|_{L^{2}\left(\Omega\right)\times L^{2}\left(\Omega% \right)}^{2}-C\,\text{meas}\left(\Omega\right)$
		$\displaystyle\geq\left(\frac{1}{2}-\frac{\sigma}{2\lambda_{1}}\right)\|u_{2}\|_{% H_{0}^{1}\times H_{0}^{1}}^{2}-C\,\text{meas}\left(\Omega\right)\rightarrow\infty,$

$\text{as \ }|u_{2}|_{H_{0}^{1}\times H_{0}^{1}}\rightarrow\infty$ , 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 $\left(H_{0}^{1}(\Omega)\right)^{2}.$

Thus, based on Theorem 4, we can state the following theorem.

Theorem 8.

Assume that (H1)-(H5) hold. Then problem (23) has a mountain pass-min solution, i.e., there is a solution $(u_{1}^{\ast},u_{2}^{\ast})\in\left(H_{0}^{1}(\Omega)\right)^{2}\times\left(H_% {0}^{1}(\Omega)\right)^{2}$ 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 pass-mountain 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 $\tilde{g}$ , instead of $g_{1}$ and $g_{2}$ (denote this condition with (H4)’), with $\tilde{g}$ defined as $\tilde{g}=g_{1}-g_{2}$ . Changing the operator $L_{2}$ results in revising the coefficients $a_{21}$ and $a_{22}$ as outlined in equations $\eqref{a_21}$ and $\eqref{a_22}$ , with $a_{21}$ being equivalent to $a_{12}$ and $a_{22}$ being equivalent to $a_{11}$ , as per equations $\eqref{a_11}$ and $\eqref{a_12}$ .

Therefore, we can state the following result.

Theorem 9.

Assume that (H1), (H2)’-(H4)’, (H5) holds true. Then problem (23) has a mountain pass-mountain pass solution, i.e., there is a solution $(u_{1}^{\ast},u_{2}^{\ast})\in\left(H_{0}^{1}(\Omega)\right)^{2}\times\left(H_% {0}^{1}(\Omega)\right)^{2}$ 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{cases}-\Delta v_{1}=a(v_{1}+w_{1})^{3}+\tilde{a}v_{1}+a(v_{1}+w_{1})% \frac{1}{v_{2}^{2}+w_{2}^{2}+1}\\ -\Delta w_{1}=a(v_{1}+w_{1})^{3}-\tilde{a}w_{1}+a(v_{1}+w_{1})\frac{1}{v_{2}^{% 2}+w_{2}^{2}+1}\\ -\Delta v_{2}=bv_{2}+\frac{1}{v_{1}^{2}+c^{2}}\\ -\Delta w_{2}=bw_{2}+\frac{1}{v_{2}^{2}+c^{2}}\end{cases}

(37)

We apply Theorem 8, where

	$\displaystyle\Omega\subset\mathbb{R}^{3},\quad a\leq\frac{\lambda_{1}}{4},\ \ % \ \tilde{a}<\frac{\lambda_{1}}{2},\quad b<1,\quad b+\tfrac{4}{c}<\lambda_{1},% \quad c>1,$
	$\displaystyle F(y_{1},z_{1},y_{2},z_{2})=\tfrac{a}{4}(y_{1}+z_{1})^{4}+\tfrac{% \tilde{a}}{2}\left(y_{1}^{2}-z_{1}^{2}\right)+\tfrac{a}{2}(y_{1}+z_{1})^{2}% \frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle G(y_{1},z_{1},y_{2},z_{2})=\tfrac{b}{2}\left(y_{2}^{2}+z_{2}^{2}% \right)+\frac{y_{2}}{y_{1}^{2}+c^{2}}+\frac{z_{2}}{z_{1}^{2}+c^{2}}.$

One can easily see that the absolute value of $F(x_{1},x_{2})$ ( $x_{1},x_{2}\in\mathbb{R}^{2}$ ) is upper-bounded by a fourth-degree polynomial in $\left|x_{1}\right|$ and

\left|G(y_{1},z_{1},y_{2},z_{2})\right|\leq\left(\tfrac{b}{2}+\tfrac{2}{c}% \right)\left|(y_{2},z_{2})\right|^{2}+\tfrac{2}{c}.

Thus condition (H1) is satisfied. Also, condition (H3) holds as $\frac{b}{2}+\frac{2}{c}<\frac{\lambda_{1}}{2}.$

Verification of the condition (H2). Since $\tfrac{(y_{1}+z_{1})^{4}}{y_{1}^{2}+z_{1}^{2}}\rightarrow 0$ provided $|y_{1}|+|z_{1}|\rightarrow 0,$ simple computations yields

\lim_{|y_{1}|+|z_{1}|\rightarrow 0}\frac{F(y_{1},z_{1},y_{2},z_{2})}{y_{1}{}^{% 2}+z_{1}{}^{2}}\leq\tfrac{\tilde{a}}{2}+a<\frac{\lambda_{1}}{2}.

On the other hand,

	$\displaystyle\lim_{\|y_{1}\|\rightarrow\infty}\frac{F((y_{1},0),x_{2})}{y_{1}{}^% {2}}$	$\displaystyle=$	$\displaystyle\lim_{\|y_{1}\|\rightarrow\infty}\frac{\tfrac{a}{4}y_{1}^{4}+\tfrac% {\tilde{a}}{2}y_{1}^{2}+\tfrac{a}{2}y_{1}^{2}\frac{1}{y_{2}^{2}+z_{2}^{2}+1}}{% y_{1}{}^{2}}$
		$\displaystyle\geq$	$\displaystyle\lim_{\|y_{1}\|\rightarrow\infty}\tfrac{a}{4}y_{1}^{2}=\infty,$

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

	$\displaystyle f_{1}(y_{1},z_{1},y_{2},z_{2})=a(y_{1}+z_{1})^{3}+\tilde{a}y_{1}% +a(y_{1}+z_{1})\frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle f_{2}(y_{1},z_{1},y_{2},z_{2})=a(y_{1}+z_{1})^{3}-\tilde{a}z_{1}% +a(y_{1}+z_{1})\frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle g_{1}(y_{1},z_{1},y_{2},z_{2})=by_{2}+\frac{1}{y_{1}^{2}+c^{2}},$
	$\displaystyle g_{2}(y_{1},z_{1},y_{2},z_{2})=bz_{2}+\frac{1}{z_{1}^{2}+c^{2}},$

which clearly gives

\widetilde{f}(y_{1},z_{1},y_{2},z_{2})=\tilde{a}y_{1}+\tilde{a}z_{1}.

Note that the function $h\colon\mathbb{R}\rightarrow\mathbb{R}$ defined as $h(x)=\tfrac{1}{x^{2}+c^{2}}$ is Lipschitz continuous, with a Lipschitz constant not greater than $\tfrac{1}{c},$ provided that $c\geq 1$ , i.e.,

\left|\frac{1}{x^{2}+c^{2}}-\frac{1}{\overline{x}^{2}+c^{2}}\right|\leq\tfrac{% 1}{c}|x-\overline{x}|,\ \text{ for all $x,\overline{x}\in\mathbb{R}$}.\text{ }

(38)

From the linearity of $\widetilde{f}$ and the Lipschitz property (38), it follows that

	$\displaystyle\left(\widetilde{f}(y_{1},z_{1},y_{2},z_{2})-\widetilde{f}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(y_% {1}-\overline{y}_{1})$
	$\displaystyle\leq\tilde{a}\|y_{1}-\overline{y}_{1}\|^{2}+\tilde{a}\left\|y_{1}-% \overline{y}_{1}\right\|\|z_{1}-\overline{z}_{1}\|,$
	$\displaystyle\left(\widetilde{f}(y_{1},z_{1},y_{2},z_{2})-\widetilde{f}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(z_% {1}-\overline{z}_{1})$
	$\displaystyle\leq\tilde{a}\|z_{1}-\overline{z}_{1}\|^{2}+\tilde{a}\left\|y_{1}-% \overline{y}_{1}\right\|\|z_{1}-\overline{z}_{1}\|,$
	$\displaystyle\left(g_{1}(y_{1},z_{1},y_{2},z_{2})-g_{1}(\overline{y}_{1},% \overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(y_{2}-\overline{y}_% {2})$
	$\displaystyle\leq b\|y_{2}-\overline{y}_{2}\|^{2}+\tfrac{1}{c}\|y_{2}-\overline{y% }_{2}\|\|y_{1}-\overline{y}_{1}\|,$
	$\displaystyle\left(g_{2}(y_{1},z_{1},y_{2},z_{2})-g(\overline{y}_{1},\overline% {z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(z_{2}-\overline{z}_{2})$
	$\displaystyle\leq b\|z_{2}-\overline{z}_{2}\|^{2}+\tfrac{1}{c}\|z_{1}-\overline{z% }_{1}\|\|z_{2}-\overline{z}_{2}\|.$

Hence the monotonicity conditions ((H4)) hold with

$\displaystyle m_{11}$	$\displaystyle=$	$\displaystyle\tilde{a},\ \ m_{12}=\tilde{a},\ \ m_{13}=0,\ \ m_{14}=0,$
$\displaystyle m_{21}$	$\displaystyle=$	$\displaystyle\tilde{a},\ \ m_{22}=\tilde{a},\ \ m_{23}=0,\ \ m_{24}=0,$
$\displaystyle m_{31}$	$\displaystyle=$	$\displaystyle\frac{1}{c},\ \ m_{32}=0,\ \ m_{33}=b,\ \ m_{34}=0,$
$\displaystyle m_{41}$	$\displaystyle=$	$\displaystyle 0,\ \ m_{42}=\frac{1}{c},\ \ m_{43}=0,\ \ m_{44}=b.$

Verification of the condition (H5). Simple computations yield

M=\begin{bmatrix}1-\beta\left(1-2\tfrac{\tilde{a}}{\lambda_{1}}\right)&0\\ \tfrac{1}{c\lambda_{1}}&b\end{bmatrix}.

Since $b<1$ and $1-2\tfrac{\tilde{a}}{\lambda_{1}}>0,$ 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 $\left(v_{1}^{\ast},w_{1}^{\ast},v_{2}^{\ast},w_{2}^{\ast}\right),$ where if $u_{1}^{\ast}:=$ $\left(v_{1}^{\ast},w_{1}^{\ast}\right)$ and $u_{2}^{\ast}:=\left(v_{2}^{\ast},w_{2}^{\ast}\right),$ one has that $u_{1}^{\ast}$ is a mountain pass type critical point of the energy functional $E_{1}\left(\cdot,u_{2}^{\ast}\right),$ and $u_{2}^{\ast}$ is a minimizer of the energy functional $E_{2}\left(u_{1}^{\ast},\cdot\right).$

Example 2. Consider the Dirichlet problem

\begin{cases}-\Delta v_{1}=a(v_{1}+w_{1})^{3}+\tilde{a}v_{1}+a(v_{1}+w_{1})% \frac{1}{v_{2}^{2}+w_{2}^{2}+1}\\ -\Delta w_{1}=a(v_{1}+w_{1})^{3}-\tilde{a}w_{1}+a(v_{1}+w_{1})\frac{1}{v_{2}^{% 2}+w_{2}^{2}+1}\\ -\Delta v_{2}=a(v_{2}+w_{2})^{3}+\tilde{a}v_{2}+a(v_{2}+w_{2})\frac{1}{v_{1}^{% 2}+w_{1}^{2}+1}\\ -\Delta w_{2}=a(v_{2}+w_{2})^{3}-\tilde{a}w_{2}+a(v_{2}+w_{2})\frac{1}{v_{1}^{% 2}+w_{1}^{2}+1}\end{cases}

(39)

We apply Theorem 9, where

	$\displaystyle\Omega\subset\mathbb{R}^{3},\quad a\leq\frac{\lambda_{1}}{4},\ \ % \ \tilde{a}<\frac{\lambda_{1}}{2},\quad$
	$\displaystyle F(y_{1},z_{1},y_{2},z_{2})=\tfrac{a}{4}(y_{1}+z_{1})^{4}+\tfrac{% \tilde{a}}{2}\left(y_{1}^{2}-z_{1}^{2}\right)+\tfrac{a}{2}(y_{1}+z_{1})^{2}% \frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle G(y_{1},z_{1},y_{2},z_{2})=\tfrac{a}{4}(y_{2}+z_{2})^{4}+\tfrac{% \tilde{a}}{2}\left(y_{2}^{2}-z_{2}^{2}\right)+\tfrac{a}{2}(y_{2}+z_{2})^{2}% \frac{1}{y_{1}^{2}+z_{1}^{2}+1}.$

Note that both $|F(x_{1},x_{2})|$ and $|G(x_{1},x_{2})|$ ( $x_{1},x_{2}\in\mathbb{R}^{2}$ ) are upper-bounded by fourth-degree 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})\leq 0,\ \text{ for all $|x_{2}|\geq R$}.

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})\leq M\leq\frac{\sigma}{2}|x_{2}|^{2}+M.

Verification of the condition (H4)’. First note that

	$\displaystyle f_{1}(y_{1},z_{1},y_{2},z_{2})=a(y_{1}+z_{1})^{3}+\tilde{a}y_{1}% +a(y_{1}+z_{1})\frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle f_{2}(y_{1},z_{1},y_{2},z_{2})=a(y_{1}+z_{1})^{3}-\tilde{a}z_{1}% +a(y_{1}+z_{1})\frac{1}{y_{2}^{2}+z_{2}^{2}+1},$
	$\displaystyle g_{1}(y_{1},z_{1},y_{2},z_{2})=a(y_{2}+z_{2})^{3}+\tilde{a}y_{2}% +a(y_{2}+z_{2})\frac{1}{y_{1}^{2}+z_{1}^{2}+1},$
	$\displaystyle g_{2}(y_{1},z_{1},y_{2},z_{2})=a(y_{2}+z_{2})^{3}-\tilde{a}z_{2}% +a(y_{2}+z_{2})\frac{1}{y_{1}^{2}+z_{1}^{2}+1},$

which gives

	$\displaystyle\widetilde{f}(y_{1},z_{1},y_{2},z_{2})=\tilde{a}y_{1}+\tilde{a}z_% {1},$
	$\displaystyle\widetilde{g}(y_{1},z_{1},y_{2},z_{2})=\tilde{a}y_{2}+\tilde{a}z_% {2}.$

The linearity of $\widetilde{f}$ and $\widetilde{g}$ yields

	$\displaystyle\left(\widetilde{f}(y_{1},z_{1},y_{2},z_{2})-\widetilde{f}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(y_% {1}-\overline{y}_{1})$
	$\displaystyle\leq\tilde{a}\|y_{1}-\overline{y}_{1}\|^{2}+\tilde{a}\left\|y_{1}-% \overline{y}_{1}\right\|\|z_{1}-\overline{z}_{1}\|,$
	$\displaystyle\left(\widetilde{f}(y_{1},z_{1},y_{2},z_{2})-\widetilde{f}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(z_% {1}-\overline{z}_{1})$
	$\displaystyle\leq\tilde{a}\|z_{1}-\overline{z}_{1}\|^{2}+\tilde{a}\left\|y_{1}-% \overline{y}_{1}\right\|\|z_{1}-\overline{z}_{1}\|,$
	$\displaystyle\left(\widetilde{g}(y_{1},z_{1},y_{2},z_{2})-\widetilde{g}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(y_% {2}-\overline{y}_{2})$
	$\displaystyle\leq\tilde{a}\|y_{2}-\overline{y}_{2}\|^{2}+\tilde{a}\|y_{2}-% \overline{y}_{2}\|\|z_{2}-\overline{z}_{2}\|,$
	$\displaystyle\left(\widetilde{g}(y_{1},z_{1},y_{2},z_{2})-\widetilde{g}(% \overline{y}_{1},\overline{z}_{1},\overline{y}_{2},\overline{z}_{2})\right)(z_% {2}-\overline{z}_{2})$
	$\displaystyle\leq\tilde{a}\|z_{2}-\overline{z}_{2}\|^{2}+\tilde{a}\|y_{2}-% \overline{y}_{2}\|\|z_{2}-\overline{z}_{2}\|.$

Hence the monotonicity conditions ((H4)) hold with

$\displaystyle m_{11}$	$\displaystyle=$	$\displaystyle\tilde{a},\ \ m_{12}=\tilde{a},\ \ m_{13}=0,\ \ m_{14}=0,$
$\displaystyle m_{21}$	$\displaystyle=$	$\displaystyle\tilde{a},\ \ m_{22}=\tilde{a},\ \ m_{23}=0,\ \ m_{24}=0,$
$\displaystyle m_{31}$	$\displaystyle=$	$\displaystyle 0,\ \ m_{32}=0,\ \ m_{33}=\tilde{a},\ \ m_{34}=\tilde{a},$
$\displaystyle m_{41}$	$\displaystyle=$	$\displaystyle 0,\ \ m_{42}=0,\ \ m_{43}=\tilde{a},\ \ m_{44}=\tilde{a}.$

Verification of the condition (H5). Simple computations yield

M=\begin{bmatrix}1-\beta\left(1-2\tfrac{\tilde{a}}{\lambda_{1}}\right)&0\\ 0&1-\beta\left(1-2\tfrac{\tilde{a}}{\lambda_{1}}\right)\end{bmatrix}.

Since $1-2\tfrac{\tilde{a}}{\lambda_{1}}>0,$ 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 $\left(v_{1}^{\ast},w_{1}^{\ast},v_{2}^{\ast},w_{2}^{\ast}\right),$ where if $u_{1}^{\ast}:=$ $\left(v_{1}^{\ast},w_{1}^{\ast}\right)$ and $u_{2}^{\ast}:=\left(v_{2}^{\ast},w_{2}^{\ast}\right),$ one has that $u_{1}^{\ast}$ is a mountain pass type critical point of the energy functional $E_{1}\left(\cdot,u_{2}^{\ast}\right),$ and $u_{2}^{\ast}$ is a mountain pass type critical point of the energy functional $E_{2}\left(u_{1}^{\ast},\cdot\right).$

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their reviews and valuable remarks, which significantly improved the paper.

References

[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973).
[2] Beldinski, M., Galewski, M.: Nash type equilibria for systems of non-potential equations. Appl. Math. Comput. 385, 125456 (2020).
[3] Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979).
[4] Boureanu, M.-M., Pucci, P., Rădulescu, V.D.: Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent. Complex Var. Elliptic Equ. 56, 755–767 (2011).
[5] Chabrowski, J.: Introduction to the theory of critical points. The mountain pass theorem. Ekeland’s variational principle. Instructional Workshop on Analysis and Geometry, Part III, Canberra (1995).
[6] Costa, D.G., Magalhaes, C.A.: Existence results for perturbations of the p -Laplacian. Nonlinear Anal. 24, 409–418 (1995).
[7] Costea, N., Csirik, M., Varga, C.: Linking-type results in nonsmooth critical point theory and applications. Set-Valued Var. Anal. 25, 333–-356 (2017).
[8] De Figueiredo, D.G.f: Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989).
[9] Filippucci, R., Pucci, P., Robert, F.: On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. (9) 91, no. 2, 156–177 (2009).
[10] Galewski, M.: On the mountain pass solutions to boundary value problems on the Sierpinski gasket. Results Math. 74, 167 (2019).
[11] Grossinho, M.R., Tersian, S.A.: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Springer, Dordrecht (2001).
[12] Jebelean, P., Moroşanu, Gh.: Mountain pass type solutions for discontinuous perturbations of the vector p-Laplacian. Nonlinear Funct. Anal. Appl. 10, no. 4, 591-611 (2005).
[13] Kassay, G., Rădulescu, V.D.: Equilibrium Problems and Applications. Academic Press (2019).
[14] Le Dret, H.: Nonlinear Elliptic Partial Differential Equations. Springer, Berlin (2018).
[15] Motreanu, D., Motreanu, V. V., Papageorgiou, N. S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014).
[16] Mugnai, D.: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. NoDEA Nonlinear Differential Equations Appl. 11, 379–391 (2004).
[17] Precup, R.: Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal. 75, 834–851 (2012).
[18] Precup, R.: Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3, no. 4, 197–207 (2014).
[19] Precup, R.: A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems. J. Math. Anal. Appl. 463, 412–431 (2018).
[20] Precup, R.: Componentwise localization of critical points for functionals defined on product spaces. Topol. Methods Nonlinear Anal. 58, 51–77 (2021).
[21] Precup, R., Stan, A.: Stationary Kirchhoff equations and systems with reaction terms. AIMS Mathematics 7, Issue 8, 15258–15281 (2022).
[22] Pucci, P., Rădulescu, V.: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 9, 3, 543–-584 (2010).
[23] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Nonlinear Partial Differential Equations. Conf. Board of Math Sci. 65, Amer. Math. Soc. (1986).
[24] Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999).
[25] Silva, E.A.B.: Existence and multiplicity of solutions for semilinear elliptic systems. Nonlinear Differ. Equ. Appl. 1, 339–363 (1994).
[26] Stan, A.: Nonlinear systems with a partial Nash type equilibrium . Studia Univ. Babeş-Bolyai Math. 66, 397–408 (2021).
[27] Struwe, M.: Variational Methods. Springer, Berlin (1990).

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	$\displaystyle\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}^{2}$	$\displaystyle\leq a_{22}\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}^{2}+a_{21}\left% \|u_{1}^{k}-u_{1}^{k+p}\right\|_{1}\,\left\|u_{2}^{k}-u_{2}^{k+p}\right\|_{2}$		(17)
		$\displaystyle\quad+\left\|\beta_{k}-\beta_{k+p}\right\|_{2}\left\|u_{2}^{k}-u_{2}% ^{k+p}\right\|_{2}.$

	$\displaystyle E_{1}(u_{1},u_{2})$	$\displaystyle=\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\int_{\Omega% }F(u_{1},u_{2})$
		$\displaystyle\geq\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\tfrac{% \mu}{2}\int_{\Omega}\|u_{1}(x)\|^{2}-C_{\mu}\int_{\Omega}\|u_{1}(x)\|^{p}dx$
		$\displaystyle\geq\tfrac{1}{2}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\tfrac{% \mu}{2\lambda_{1}}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-C_{\mu}^{\prime}\|u_{% 1}\|_{H_{0}^{1}\times H_{0}^{1}}^{p}$
		$\displaystyle=\left(\tfrac{1}{2}-\tfrac{\mu}{2\lambda_{1}}\right)\|u_{1}\|_{H_{0% }^{1}\times H_{0}^{1}}^{2}-C_{\theta}^{\prime}\|u_{1}\|_{H_{0}^{1}\times H_{0}^{% 1}}^{p}.$

	$\displaystyle\left(N_{1}(u_{1},u_{2})-N_{1}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{1}-\overline{u}_{1}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq\left(1-\beta\right)\left(\|v_{1}-\overline{u}_{1}\|_{H_{0}^{1}% }^{2}+\|w_{1}-\overline{w}_{1}\|_{H_{0}^{1}}^{2}\right)$
	$\displaystyle\quad+\beta m_{11}\|v_{1}-\overline{v}_{1}\|_{L^{2}}^{2}+\beta m_{1% 2}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{22}\|w_{1}-\overline{w}_{1}\|_{L^{2}}^{2}+\beta m_{2% 1}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{13}\left\|v_{1}-\overline{v}_{1}\right\|_{L^{2}}% \left\|v_{2}-\overline{v}_{2}\right\|_{L^{2}}+m_{14}\left\|v_{1}-\overline{v}_{1}% \right\|_{L^{2}}\left\|w_{2}-\overline{w}_{2}\right\|_{L^{2}}$
	$\displaystyle\quad+\beta m_{23}\left\|w_{1}-\overline{w}_{1}\right\|_{L^{2}}% \left\|v_{2}-\overline{v}_{2}\right\|_{L^{2}}+m_{24}\left\|w_{1}-\overline{w}_{1}% \right\|_{L^{2}}\left\|w_{2}-\overline{w}_{2}\right\|_{L^{2}}.$

	$\displaystyle\left(N_{2}(u_{1},u_{2})-N_{2}(\overline{u}_{1},\overline{u}_{2})% \,,\,u_{2}-\overline{u}_{2}\right)_{H_{0}^{1}\times H_{0}^{1}}$
	$\displaystyle\leq m_{33}\|v_{2}-\overline{v}_{2}\|_{H_{0}^{1}}^{2}+m_{44}\|w_{2}-% \overline{w}_{2}\|_{H_{0}^{1}}^{2}$
	$\displaystyle\quad+\left(m_{34}+m_{43}\right)\|v_{2}-\overline{v}_{2}\|_{L^{2}}\|% w_{2}-\overline{w}_{2}\|_{L^{2}}$
	$\displaystyle\quad+\|v_{1}-\overline{v}_{1}\|_{L^{2}}\left(m_{31}\|v_{1}-% \overline{v}_{1}\|_{L^{2}}+m_{32}\|w_{1}-\overline{w}_{2}\|_{L^{2}}\right)$
	$\displaystyle\quad+\|v_{2}-\overline{v}_{2}\|_{L^{2}}\left(m_{41}\|v_{1}-% \overline{v}_{1}\|_{L^{2}}+m_{42}\|w_{1}-\overline{w}_{2}\|_{L^{2}}\right),$

	$\displaystyle E_{2}\left(u_{1},u_{2}\right)$	$\displaystyle=\tfrac{1}{2}\|u_{2}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\int_{\Omega% }G(u_{1},u_{2})$
		$\displaystyle\geq\tfrac{1}{2}\|u_{2}\|_{H_{0}^{1}\times H_{0}^{1}}^{2}-\frac{% \sigma}{2}\left\|u_{2}\right\|_{L^{2}\left(\Omega\right)\times L^{2}\left(\Omega% \right)}^{2}-C\,\text{meas}\left(\Omega\right)$
		$\displaystyle\geq\left(\frac{1}{2}-\frac{\sigma}{2\lambda_{1}}\right)\|u_{2}\|_{% H_{0}^{1}\times H_{0}^{1}}^{2}-C\,\text{meas}\left(\Omega\right)\rightarrow\infty,$

	$\displaystyle\left\|F(x_{1},x_{2})\right\|$	$\displaystyle\leq$	$\displaystyle C_{F}\left(\left\|x_{1}\right\|^{p}+1\right),$		(24)
	$\displaystyle\left\|G(x_{1},x_{2})\right\|$	$\displaystyle\leq$	$\displaystyle C_{G}\left(\left\|x_{2}\right\|^{p}+1\right)\,,$

Linking Methods for Componentwise Variational Systems

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Linking methods for componentwise variational systems

Abstract

keywords:

1 Introduction and Preliminaries

1.1 A unifying notion of linking

Definition 1 ((20)).

1.2 Ekeland variational principle

Lemma 1 (Ekeland Principle - weak form).

1.3 Two auxiliary results

Lemma 2.

Proof.

Lemma 3.

2 Main results

Theorem 4.

Proof.

Theorem 5.

Proof.

Remark 1.

Theorem 6.

Proof.

Remark 2.

Remark 3.

Theorem 7.

Proof.

Remark 4.

3 Application

Theorem 8.

Theorem 9.

Acknowledgements

References

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