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Journal of Nonlinear and Convex Analysis

Volume 17, Number 8, 2016, 1–

VARIATIONAL PROPERTIES OF THE SOLUTIONS OF

SEMILINEAR EQUATIONS UNDER NONRESONANCE

CONDITIONS

ANGELA BUDESCU∗AND RADU PRECUP†

Abstract. The paper deals with weak solutions of the semilinear operator equa-

tion Au −cu =J′(u) in a Hilbert space, where Ais a positively deﬁned linear

operator, Jis a C1functional and cis not an eigenvalue of A. Under some

assumptions on J, if Eis the energy functional of the equation and clies be-

tween two eigenvalues λkand λk+1,then for any solution uof the equation,

E(u)≤E(u+w) for every element worthogonal on the ﬁrst keigenvectors of A.

The proof is based on the application of Ekeland’s variational principle to a suit-

able modiﬁed functional, and diﬀers essentially from the proof of the particular

case when c= 0.The theory is applicable to elliptic problems.

1. Introduction

Recall that the classical Dirichlet’s principle for Poisson’s equation (see, e.g.,

[2, 11]) states that the solution of the Dirichlet boundary value problem can be

characterized as the minimizer of the associated energy functional. More exactly, a

function u∈H1

0(Ω) is the weak solution of the problem

−∆u=hin Ω

u= 0 on ∂Ω,

where h∈L2(Ω) ,if and only if it minimizes on H1

0(Ω) ,the functional

E(w) = Ω1

2|∇w|2−hwdx.

An analogue result for the semilinear case of the equation −∆u=f(x, u) was

recently established in [3], even in the abstract more general setting of operator

equations of the form

(1.1) Au =J′(u),

where Ais a positively deﬁned linear operator and J′is the Fr´echet derivative of a

functional J, as an application of the corresponding theory for ﬁxed point equations

worked out in [12]. In addition, in [12, 3], the case of a system of two equations

2010 Mathematics Subject Classiﬁcation. 34G20, 47J05, 47J30, 35J20.

Key words and phrases. Semilinear operator equation, ﬁxed point, critical point, eigenvalues,

nonresonance, minimizer, Ekeland’s variational principle, elliptic problem.

∗The ﬁrst author was supported by the Sectorial Operational Programme for Human Re-

sources Development 2007-2013, coﬁnanced by the European Social Fund, under the project POS-

DRU/159/1.5/S/137750 - “Doctoral and postdoctoral programs - support for increasing research

competitiveness in the ﬁeld of exact sciences”.

†The second author was supported by a grant of the Romanian National Authority for Scientiﬁc

Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.

2 A. BUDESCU AND R. PRECUP

in variables u, v was considered, where the solution (u, v) appeared as a Nash-type

equilibrium of the pair of energy functionals associated to the two equations of

the system. For diﬀerent approaches of variational properties of the solutions of

nonlinear operator equations, we refer to [1] and [13].

The aim of this paper is to investigate the case of the equation

Au −cu =J′(u),

under the nonresonance condition c̸=λj, j = 1,2, ..., where λjare the eigenvalues

of A. The case is truly interesting as we can simply show on the linear problem

−∆u−cu =hin Ω

u= 0 on ∂Ω,

where Ω is a bounded open set in Rn, h ∈L2(Ω) and uis sought in H1

0(Ω) .It is

a standard fact that the problem has a unique weak solution, and that the energy

functional is

E(w) = Ω1

2|∇w|2−c

2w2−hwdx.

However, if c > λ1, E is not bounded from below and consequently the solution can

not be a minimizer of E. Indeed, if ϕ1is an eigenfunction of −∆ corresponding to

λ1,with ||ϕ1||L2(Ω) = 1,then

E(tϕ1) = 1

2t2(λ1−c)−tΩ

hϕ1dx → −∞ as t→ ∞.

Nevertheless, even in this case, a variational property holds for the solution. More

exactly, if we denote by ϕjthe eigenfunction corresponding to λj,we assume that

λk< c < λk+1

for some k∈ {0,1, ...},where λ0=−∞,and we denote H0=H1

0(Ω) ,

Hk=v∈H1

0(Ω) : vand ϕjare orthogonal for j= 1,2, ..., k,

then

E(u)≤E(u+v) for all v∈Hk.

We show this using the characterization

λk+1 = inf Ω|∇v|2dx

Ωv2dx :v∈Hk\ {0}

(see, e.g., [2, 11]), from which

Ω|∇v|2dx ≥λk+1 Ω

v2dx for all v∈Hk.

Then, with the notations

(u, v)H1

0(Ω) =Ω∇u· ∇vdx, (u, v)L2(Ω) =Ω

uvdx,

||u||H1

0(Ω) =Ω|∇u|2dx1/2

,||u||L2(Ω) =Ω

u2dx1/2

,

VARIATIONAL PROPERTIES OF THE SOLUTIONS 3

direct computation gives

E(u+v) = E(u)+(u, v)H1

0(Ω) −c(u, v)L2(Ω) −(h, v)L2(Ω)

+1

2||v||2

H1

0(Ω) −c

2||v||2

L2(Ω)

=E(u) + 1

2||v||2

H1

0(Ω) −c

2||v||2

L2(Ω)

≥E(u) + 1

2(λk+1 −c)||v||2

L2(Ω) ≥E(u),

the desired result.

It is our goal to extend the previous property from the linear case to the semilinear

one. The results are obtained in the abstract setting of the theory of operator

equations and complement the existence theory for such type of equations [7, 8],

with applications to elliptic problems [4, 9].

2. Preliminaries

In this section, following [5, 6], we present some basic results from the abstract

theory of linear operator equations. Let Hbe a Hilbert space with the inner prod-

uct denoted by (., .)Hand the norm ∥.∥H.Let A:D(A)⊂H→Hbe a linear

operator with D(A) dense in H, positively deﬁned, i.e., self-adjoint in the sense

that (Au, v)H= (u, Av)Hfor every u, v ∈D(A) and with

(2.1) (Au, u)H≥γ2∥u∥2

H,

for all u∈D(A) and some γ > 0.The linear subspace D(A) of His endowed with

the inner product

(u, v)HA:= (Au, v)H

and the energetic norm ∥u∥HA=(Au, u)H.The completion of (D(A),(., .)HA)

is called the energetic space of Aand is denoted by HA.We use the same symbols

(., .)HAand ∥.∥HAto denote the induced inner product and norm on the larger space

HA.From (2.1), by density, we deduce the Poincar´e-type inequality for the inclusion

HA⊂H,

(2.2) ∥u∥H≤1

γ∥u∥HA(u∈HA).

It can be used in order to identify the elements of HAwith elements from H. Let

H′

Abe the dual space of HA.If we identify Hwith its dual, then from HA⊂Hwe

have H⊂H′

A.

From the Riesz representation theorem, it follows that for each f∈H′

A,there

exists a unique uf∈HAsuch that

(2.3) (uf, v)HA= (f, v) for every v∈HA,

where the notation (f, v) stands for the value of the functional fon the element v.

We denote ufby A−1fand we call it the weak solution of the equation Au =f.

Thus

(2.4) A−1:H′

A→HA,A−1f, vHA= (f, v) for f∈H′

A, v ∈HA.

4 A. BUDESCU AND R. PRECUP

From (2.4) and the deﬁnition of the norm of a continuous linear functional, we can

deduce that the operator A−1is an isometry between H′

Aand HA,i.e,

(2.5) ∥A−1f∥HA=∥f∥H′

A,

for all f∈H′

A.

Note that a Poincar´e-type inequality also holds for the inclusion H⊂H′

A.Indeed,

if u∈H, then using (2.5), (2.4) and (2.2) one has

∥u∥2

H′

A=

A−1u

2

HA=u, A−1uH≤ ∥u∥H

A−1u

H

≤1

γ∥u∥H

A−1u

HA=1

γ∥u∥H∥u∥H′

A.

Hence

(2.6) ∥u∥H′

A≤1

γ∥u∥H(u∈H).

It is easy to see that the linear operator A−1from Hto His positively deﬁned.

From now on, in addition, we shall assume that the embedding of HAinto His

compact. This guarantees that A−1is a compact operator from Hto itself. Then,

from the spectral theory of self-adjoint compact operators (see [6, 8]), we know

that all eigenvalues of A−1are positive; the set of eigenvalues of A−1is nonempty

and at most countable; zero is the only possible cluster point of it; there exists

an orthonormal set (ϕj) of eigenvectors of A−1, with ∥ϕj∥H= 1,which is at most

countable and it is complete in the image of A−1, i.e.,

A−1u=A−1u, ϕjHϕjfor all u∈H.

Assume that D(A) is inﬁnite dimensional. Then the image of A−1is inﬁnite di-

mensional and so there exists a sequence (µj)j≥1of eigenvalues of A−1and corre-

spondingly, a sequence (ϕj)j≥1of eigenvalues, orthonormal in H. Let λj:= 1/µj.

Then 0 < λ1≤λ2≤... ≤λj≤..., λj→ ∞ as j→ ∞,and from A−1ϕj=µjϕj,we

have

(2.7) (ϕj, v)HA=λj(ϕj, v)Hfor all v∈HA,

i.e., Aϕj=λjϕjin the weak sense. Hence λjand ϕj, j ≥1,are the eigenvalues and

eigenvectors of A, with ∥ϕj∥H= 1.Also, we shall use the following characterization

for the eigenvalues (see [5, 6]), namely

λj= inf ∥u∥2

HA

∥u∥2

H

:u∈HA\ {0},(u, ϕi)HA= 0 for i= 1,2, ..., j −1

(j= 1,2, ...).Note that for j= 1,this shows that the best constant γ2in (2.1) is

γ2=λ1.

If in H′

Awe consider the inner product and norm

(u, v)H′

A:= A−1u, A−1vHA,∥u∥H′

A:=

A−1u

HA,

then using (2.7) we obtain:

∥ϕj∥H= 1,∥ϕj∥HA=λj,∥ϕj∥H′

A=1

λj

.

VARIATIONAL PROPERTIES OF THE SOLUTIONS 5

Also, for each v∈H′

A,one has

(v, ϕj) = (A−1v, ϕj)HA= (A−1v, A−1(λjϕj))HA=λj(v, ϕj)H′

A.

This implies that the systems

(ϕj),1

λj

ϕj,λjϕj

are orthonormal and complete (Hilbert bases) in H, HAand H′

A,respectively, and

that for each v∈HA,the Fourier series

(v, ϕj)Hϕj,v, 1

λj

ϕjHA

1

λj

ϕj,v, λjϕjH′

Aλjϕj

are identical and can be written as

(v, ϕj)ϕj,

where by (v, ϕj) we mean the action of vas an element of H′

Aover ϕj.

In order to make clear the conclusion of our main result, we conclude this section

by reminding one of the results from [3] with regard to equation (1.1) and its

corresponding energy functional E:HA→R,

E(u) = 1

2∥u∥2

HA−J(u).

Theorem 2.1. Let Abe a linear operator as above and J:H→Rbe a C1

functional. In addition assume that there exist α < λ1,a≤1

2and b∈R+such

that the following conditions hold:

(2.8) ∥J′(u)−J′(v)∥H≤α∥u−v∥H

for all u, v ∈H, and

(2.9) J(u)≤a∥u∥2

HA+b

for all u∈HA.Then the equation (1.1) has a unique weak solution u∗∈HAand

the following variational property holds

E(u∗) = inf

v∈HA

E(v).

3. The main result

The main result of this paper is concerning with a variational property of the

solutions of semilinear equations of the form

(3.1) Au −cu =J′(u),

where Ais a linear operator having all the properties required in Section 2 and cis

not an eigenvalue of A. We shall work out a general theory of nonresonance which

in particular, for c= 0,contains the results from [3].

Let J:H→Rbe a C1functional and ca constant such that

λk< c < λk+1

6 A. BUDESCU AND R. PRECUP

for some k∈ {0,1, ...},where by λ0we mean −∞.We underline that the index

kwill be essential in what follows. We look for weak solutions u∈HAto the

semilinear equation (3.1), i.e., an element u∈HAwith

(3.2) (u, v)HA−c(u, v)H= (J′(u), v) for all v∈HA.

If we denote

Lu =Au −cu,

then (3.1) is equivalent to the ﬁxed point equation

u=L−1J′(u), u ∈HA.

On the other hand, the equation (3.1) has the variational form E′(u)=0,where

E:HA→Ris the energy functional

E(u) = 1

2∥u∥2

HA−c

2∥u∥2

H−J(u).

Note that

E′(u) = Lu −J′(u).

If we identify H′

Awith HAvia A−1and we take into account that

A−1(Lu −J′(u)) = A−1(Au −cu −J′(u)) = u−A−1[cu +J′(u)],

we obtain

(3.3) E′(u) = u−A−1[cu +J′(u)].

We remark that the method we used in [3] can not longer be applied when c̸= 0,

as we can see that

A−1[cu +J′(u)] ̸=L−1J′(u).

Let Hkand H⊥

kbe the subspaces of HAdeﬁned by

(3.4) Hk=u∈HA: (u, ϕj)HA= 0 for j= 1,2, ..., k;

(3.5) H⊥

k=u∈HA: (u, ϕj)HA= 0 for j=k+ 1, k + 2, ....

In what follows, by Pand P⊥we mean the projection operators on Hkand on its

orthogonal complement H⊥

k. So any element u∈HAcan be written as

(3.6) u=P⊥u+P u.

The main result of this paper is the following theorem.

Theorem 3.1. Assume that all the above conditions on A, J and chold. In addition

assume that there exist α < λk+1 −c, p ≤1

2−c

2λk+1 and q, r ∈R+such that

(3.7)

J′(u)−J′(v)

H≤α∥u−v∥H

for all u, v ∈HAsatisfying P⊥u=P⊥v, and

(3.8) J(u)≤p∥u∥2

HA+q∥u∥HA+r

for all u∈HA.Then for any weak solution u∗∈HAof the equation (3.1), the

following variational property holds

(3.9) E(u∗) = inf

w∈Hk

E(u∗+w).

VARIATIONAL PROPERTIES OF THE SOLUTIONS 7

4. Proof of the main result

We start with some auxiliary results. The ﬁrst lemma extends to H′

Athe corre-

sponding result for Hused in [7, 8] and ﬁrst proved for A=−∆ in [10] (see also [9,

Lemma 6.1]).

Lemma 4.1. Let cbe any constant with c̸=λj,for j= 1,2, ... . For each v∈H′

A,

there exists a unique weak solution u∈HAto the problem

(4.1) Lu := Au −cu =v, u ∈HA,

denoted by L−1v, and the following eigenvector expansion holds

(4.2) L−1v=

∞

j=1

1

λj−c(v, ϕj)ϕj,

where the series converges in HA.In addition

(4.3) ∥L−1v∥HA≤σc∥v∥H′

A,

where σc= max

λj

λj−c

:j= 1,2, ....

Proof. We ﬁrst prove the convergence of the series (4.2) in HA.Since (λ−1/2

jϕj) is

a Hilbert base in HA,we have

n+p

j=n+1

1

λj−c(v, ϕj)ϕj

2

HA

=

n+p

j=n+1

λj

λj−cv, ϕj

λjϕj

λj

2

HA

(4.4)

=

n+p

j=n+1 λj

λj−c2v, ϕj

λj2

=

n+p

j=n+1 λj

λj−c21

λj

(v, ϕj)2.

Furthermore using (v, ϕj) = λj(v, ϕj)H′

Awe deduce

n+p

j=n+1 λj

λj−c21

λj

(v, ϕj)2=

n+p

j=n+1 λj

λj−c2

λj(v, ϕj)2

H′

A

(4.5)

=

n+p

j=n+1 λj

λj−c2v, λjϕj2

H′

A

≤σ2

c

n+p

j=n+1 v, λjϕj2

H′

A

.

Since λjϕjis a Hilbert base in H′

A,according to Parseval’s identity, the last

sum is less than any ε > 0 for large enough n. Thus the sequence of partial sums

8 A. BUDESCU AND R. PRECUP

of the series (4.2) is Cauchy and so convergent in HA.Let u∈HAbe the sum of

series (4.2). Next we show that Lu =vweakly, i.e.,

(u, w)HA−c(u, w)H= (v, w) for all w∈HA.

Indeed, one has

(u, w)HA=

∞

j=1

1

λj−c(v, ϕj) (w, ϕj)HA=

∞

j=1

λj

λj−c(v, ϕj) (w, ϕj)H

and

(u, w)H=

∞

j=1

1

λj−c(v, ϕj) (w, ϕj)H.

Hence

(u, w)HA−c(u, w)H=

∞

j=1

(v, ϕj) (w, ϕj)H=∞

j=1

(v, ϕj)ϕj, w= (v , w)

as desired. The uniqueness of the solution ufollows immediately from c̸=λj,

j= 1,2, .... Finally, (4.3) follows from (4.4), (4.5) if we take n= 0 and we let

p→ ∞.□

Remark 4.2. By considering the cases j < k and j > k + 1,where λk< c

< λk+1,it is easy to show that

σc= max λk

c−λk

,λk+1

λk+1 −c.

The next result is about the subspace Hkgiven by (3.4).

Lemma 4.3. For every w∈Hk,the following inequalities hold:

(4.6) ∥w∥H≤1

λk+1 ∥w∥HA;

(4.7)

L−1w

HA≤λk+1

λk+1 −c∥w∥H;

(4.8)

L−1w

HA≤1

λk+1 −c∥w∥HA.

Proof. Since w∈Hk,one has

w=

∞

j=k+1

(w, ϕj)Hϕj.

Then

∥w∥2

H=

∞

j=k+1

(w, ϕj)2

H=

∞

j=k+1 w, A−1ϕj2

HA=

∞

j=k+1 w, 1

λj

ϕj2

HA

=

∞

j=k+1

1

λjw, ϕj

λj2

HA

≤1

λk+1

∞

j=k+1 w, ϕj

λj2

HA

VARIATIONAL PROPERTIES OF THE SOLUTIONS 9

=1

λk+1 ∥w∥2

HA.

Hence (4.6) is proved. Furthermore, one has

L−1w=

∞

j=k+1

1

λj−c(w, ϕj)ϕj.

Then

L−1w

2

HA=

∞

j=k+1

1

λj−cA−1w, ϕjHAϕj

2

HA

=

∞

j=k+1

λj

λj−cA−1w, ϕj

λjHA

ϕj

λj

2

HA

=

∞

j=k+1 λj

λj−c2A−1w, ϕj

λj2

HA

=

∞

j=k+1

λj

(λj−c)2(w, ϕj)2

H.

It is easy to check that for j≥k+ 1,

λj

(λj−c)2≤λk+1

(λk+1 −c)2.

It follows that

L−1w

2

HA≤λk+1

(λk+1 −c)2

∞

j=k+1

(w, ϕj)2

H=λk+1

(λk+1 −c)2∥w∥2

H,

which is (4.7). Finally, (4.8) is a direct consequence of (4.6) and (4.7). □

Proof of Theorem 3.1. To any ﬁxed element u∈HA,we associate the functional

Ek:HA→R,given by

Ek(v) = EP⊥u+P v.

Clearly, the functional Ekdepends on P⊥u. The choice of the element uwill be

made at the end of the proof.

Step 1: The functional Ekis bounded from below. Indeed, for each v∈HA,one

has P v ∈Hkand so, by the deﬁnition of λk+1,∥P v∥2

HA≥λk+1 ∥P v∥2

H.Also from

(3.8), since P⊥uand P v are orthogonal, we have

JP⊥u+P v≤p

P⊥u+P v

2

HA

+q

P⊥u+P v

HA

+r

≤p∥P v∥2

HA+q∥P v∥HA+r,

where r=r+p

P⊥u

2

HA+q

P⊥u

HA.Then

Ek(v) = 1

2

P⊥u+P v

2

HA−c

2

P⊥u+P v

2

H−JP⊥u+P v

10 A. BUDESCU AND R. PRECUP

=1

2

P⊥u

2

HA

+1

2∥P v∥2

HA−c

2

P⊥u

2

HA

−c

2∥P v∥2

H−JP⊥u+P v

≥1

2∥P v∥2

HA−c

2∥P v∥2

H+1

2

P⊥u

2

HA−c

2

P⊥u

2

H

−p∥P v∥2

HA+q∥P v∥HA+r

≥1

2−c

2λk+1 −p∥P v∥2

HA−q∥P v∥HA−r1≥ −b1.

Here r1=r−1

2

P⊥u

2

HA+c

2

P⊥u

2

Hand b1is the minimum value of the

quadratic function 1

2−c

2λk+1 −pt2−qt −r1.Hence Ekis bounded from below.

Notice that the bound −b1depends on the ﬁxed element u.

Step 2: Ekis a C1functional. To show this, for any v, w ∈HA,we compute the

directional derivative

E′

k(v), w= lim

t→0

Ek(v+tw)−Ek(v)

t

= lim

t→0

EP⊥u+P v +tP w−EP⊥u+P v

t.

Routine calculation and (3.3) give

E′

k(v), w

=E′P⊥u+P v, P w

=P⊥u+P v −A−1cP⊥u+P v+J′P⊥u+P v , P wHA

.

Furthermore, using the orthogonality between Hkand H⊥

k,and the properties

A−1(Hk)⊂Hkand A−1H⊥

k⊂H⊥

k,we deduce that

E′

k(v), w

=P v −A−1cP v +J′P⊥u+P v , P wHA

=P v −A−1cP v +P⊥J′P⊥u+P v +P J′P⊥u+P v , P w HA

=P v −A−1cP v +P J ′P⊥u+P v, P wHA

=P v −A−1cP v +P J ′P⊥u+P v, P ⊥w+P wHA

=P v −A−1cP v +P J ′P⊥u+P v, wHA

.

Hence

(4.9) E′

k(v) = P v −A−1cP v +P J ′P⊥u+P v for every v∈HA.

Thus Ekis a C1functional, bounded from below. Then Ekeland’s variational prin-

ciple (see, e.g., [14, Corollary 5.3]) guarantees the existence of a sequence (vj) in

VARIATIONAL PROPERTIES OF THE SOLUTIONS 11

HAsuch that

(4.10) Ek(vj)→inf

v∈HA

Ek(v), E′

k(vj)→0 as j→ ∞.

Step 3: The sequence (P vj) is convergent in HA.To prove this, we ﬁrst see that

if in (4.9) we apply A, then

AE′

k(v) = AP v −cP v −P J ′P⊥u+P v,

that is

LP v =AE′

k(v) + P J ′P⊥u+P v,

which for v=vj,gives

(4.11) P vj=L−1AE′

k(vj) + L−1P J ′P⊥u+P vj.

Denote wj:= L−1AE′

k(vj).Since E′

k(vj)→0 as j→ ∞,one can show that

wj→0 as j→ ∞ too. Next, from (4.11) one has

∥P vn+p−P vn∥HA

≤ ∥wn+p−wn∥HA

+

L−1PJ′P⊥u+P vn+p−J′P⊥u+P vn

HA

.

Taking into account (4.7) and the fact that Pis nonexpansive, we obtain

L−1PJ′P⊥u+P vn+p−J′P⊥u+P vn

HA

≤λk+1

λk+1 −c

PJ′P⊥u+P vn+p−J′P⊥u+P vn

H

≤λk+1

λk+1 −c

J′P⊥u+P vn+p−J′P⊥u+P vn

H.

Furthermore, the hypothesis (3.7) and Poincar´e’s inequality for the inclusion HA⊂

Hyield

L−1PJ′P⊥u+P vn+p−J′P⊥u+P vn

HA

≤λk+1

λk+1 −cα∥P vn+p−P vn∥H≤α

λk+1 −c∥P vn+p−P vn∥HA.(4.12)

As a result

∥P vn+p−P vn∥HA≤ ∥wn+p−wn∥HA+α

λk+1 −c∥P vn+p−P vn∥HA,

whence

∥P vn+p−P vn∥HA≤1

1−α

λk+1−c∥wn+p−wn∥HA.

The sequence (wj) being convergent, this shows that the sequence (P vj) is Cauchy

in HA.

12 A. BUDESCU AND R. PRECUP

Step 4: Properties of the limit of the sequence (P vj).Let v∗=v∗(u) be the limit

of (P vj).Clearly v∗∈Hk,and so P v∗=v∗.Now, we pass to the limit in (4.10).

First, from

Ek(vj) = EP⊥u+P vjand Ek(vj)→inf

v∈HA

Ek(v)

we obtain

EP⊥u+v∗= inf

v∈HA

Ek(v).

Since P v∗=v∗,we have EP⊥u+v∗=EP⊥u+P v∗=Ek(v∗).Hence

(4.13) Ek(v∗) = inf

v∈HK

Ek(v).

Next, passing to the limit in

E′

k(vj) = P vj−A−1cP vj+P J ′P⊥u+P vj

and using the second conclusion from Ekeland’s variational principle, we obtain

E′

k(v∗) = 0.

Step 5: The projection on Hkof the set of all critical points of the functional Ek

contains a single point. Indeed, for any critical point v∈HAof Ek,one has

P v =A−1cP v +P J ′P⊥u+P v,

whence applying Awe obtain

AP v =cP v +P J ′P⊥u+P v,

that is

LP v =P J ′P⊥u+P v.

Thus, the equation E′

k(v) = 0 is equivalent to

P v =L−1P J ′P⊥u+P v.

This shows that P v is a ﬁxed point of the operator M:Hk→Hkgiven by

Mw := L−1P J ′P⊥u+w, w ∈Hk.

It is easy to see that in view of the relation (4.12) and of Lemma 4.3, the operator

Mis a contraction on Hk.Hence, by Banach’s contraction principle, the operator

Mhas a unique ﬁxed point. Consequently, the projection on Hkof any critical

point vof Ekis the unique ﬁxed point of the operator M.

Step 6: If u∗is a solution of the equation (3.1), then u∗is a critical point of

the functional Ekassociated to u:= u∗.To prove this, let u∗be a solution of the

equation (3.1). Then u∗=A−1[cu∗+J′(u∗)] and if we apply P, we have

P u∗=P A−1cP ⊥u∗+cP u∗+P⊥J′(u∗) + P J ′(u∗)

=P A−1cP u∗+P J ′(u∗).

VARIATIONAL PROPERTIES OF THE SOLUTIONS 13

This together with A−1[cP u∗+P J ′(u∗)] ⊂Hkshows that

P u∗=A−1cP u∗+P J ′(u∗)=A−1cP u∗+P J′P⊥u∗+P u∗.

In view of (4.9), this proves that u∗is a critical point of the functional Ekassociated

to u:= u∗, i.e., E′

k(u∗) = 0.

Step 7: Proof of (3.9). From Step 5 we know that the projection P u∗of u∗is

uniquely determined. Hence

P v∗(u∗) = P u∗,

where v∗=v∗(u∗) is the limit of the sequence (P vj) from Step 4. Since v∗∈Hk,

one has P v∗(u∗) = v∗(u∗).Consequently,

v∗(u∗) = P u∗.

We remark that

(4.14) E(u∗) = Ek(v∗),

which follows from

E(u∗) = EP⊥u∗+P u∗

and

Ek(v∗) = EP⊥u∗+P v∗=EP⊥u∗+P u∗.

Furthermore, using (4.13) we can see that

Ek(v∗) = inf

v∈HA

Ek(v) = inf

v∈HA

EP⊥u∗+P v= inf

v∈Hk

EP⊥u∗+v.

Since any v∈Hkcan be written as v=P u∗+wwith w∈Hk,we obtain

(4.15) Ek(v∗) = inf

w∈Hk

EP⊥u∗+P u∗+w= inf

w∈Hk

E(u∗+w).

Now (4.14) and (4.15) yield (3.9), and the proof is complete. □

Remark 4.4. If in Theorem 3.1, we assume more, namely that the inequality (3.7)

is satisﬁed for all u, v ∈HA,and that α < λ1/σc,then the equation (3.1) has a

unique solution. Indeed, in this case

∥L−1J′(u)−L−1J′(v)∥HA≤σc∥J′(u)−J′(v)∥H′

A

≤σc

√λ1∥J′(u)−J′(v)∥H≤ασc

√λ1∥u−v∥H≤ασc

λ1∥u−v∥HA,

which shows that the operator L−1J′is a contraction on HA.Also note that if

c= 0,then σc= 1,the two conditions on α, α < λk+1 −cand α < λ1

σccoincide, and

Theorem 3.1 reduces to the corresponding result from [3].

As in paper [3], the abstract theory presented here can be applied to semilinear

elliptic equations under nonresonance conditions.

In a forthcoming work, motivated by our previous papers [3, 12], we shall extend

the theory to the nonresonance systems of the type

(4.16) A1u−c1u=J11 (u, v)

A2v−c2v=J22 (u, v),

14 A. BUDESCU AND R. PRECUP

where by J11(u, v), J22 (u, v) we mean the partial derivatives of two C1functionals

J1, J2:H1×H2→R.We can anticipate that under suitable conditions, the solution

(u, v) of the system is a Nash-type equilibrium of the pair of associated energy

functionals on a suitable cross product subspace depending on indices k1and k2,

where λki< ci< λki+1, i = 1,2.

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Manuscript received ,

revised ,

A. Budescu

Babe¸s-Bolyai University, Department of Mathematics, 400084 Cluj, Romania

E-mail address:Budescu.Angela@math.ubbcluj.ro

R. Precup

Babe¸s-Bolyai University, Department of Mathematics, 400084 Cluj, Romania

E-mail address:r.precup@math.ubbcluj.ro