Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 10, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A GRADIENT

TYPE COOPERATIVE/COMPETITIVE ELLIPTIC SYSTEM

KAYE SILVA, STEFFÂNIO MORENO SOUSA

Abstract. We study the existence of positive solutions for gradient type co-

operative, competitive elliptic systems, which depends on real parameters λ, µ.

Our analysis is purely variational and depends on finer estimates with respect
to the Nehari sets, in fact, we determine the extremal parameter λ∗(µ) for

which the Nehari set is a manifold and hence standard variational techniques

can be applied. We also discuss the cases where the Nehari set is not a mani-
fold.

1. Introduction

In this work we study the gradient type cooperative or competitive elliptic system

−∆u = µu+ λv + f(x)|u|p−2u in Ω,

−∆v = λu+ µv − g(x)|v|q−2v in Ω,

u = v = 0 on ∂Ω,

(1.1)

where λ, µ are real parameters, Ω is a bounded domain in RN with smooth boundary
∂Ω, N ≥ 1, 2 < q < p < 2∗, 2∗ = 2N/(N − 2). We look for weak solutions in
X := H1

0 (Ω)×H1
0 (Ω).

Gradient type systems and cooperative or competitive type systems have been
studied by many authors: see for example the works of deFigueiredo [8], Clément
et al. [5], Alves et al. [1], Bozhkov and Mitidieri [3], Wenming [14], Costa and
Magalhães [6], da Silva [7] and the references therein. Such systems appear in
many phenomena in Physics, Chemistry, Biology, etc. (see for example Brown
[4] and the references therein), in particular they are related to reaction-diffusion
systems that appear in chemical and biological phenomena.

Our plan is to study system (1.1) with respect to the parameters λ, µ only by a
variational method. We will provide a relation between the parameters λ, µ with
some topological properties of the Nehari set and the existence of solutions to
problem (1.1). From now on, a solution to (1.1) is a critical point to the energy
functional Φλ,µ : X → R which is defined by

Φλ,µ(u, v) =
1

2

(∫
|∇u|2 +

∫
|∇v|2

)
− µ

2

(∫
|u|2 +

∫
|v|2
)

2010 Mathematics Subject Classification. 35A02, 35A15, 35B32.
Key words and phrases. Gradient systems; cooperative/competitive systems;

Nehari manifold; variational methods; extremal parameter.
c©2020 Texas State University.

Submitted December 21, 2019. Published January 23, 2020.

1



2 K. SILVA, S. M. SOUSA EJDE-2020/10

− λ
∫
uv +

1

q

∫
g|v|q − 1

p

∫
f |u|p.

Depending on the values of the parameters λ, µ, the energy functional Φλ,µ is
unbounded from below and from above. This kind of behavior is similar to that
of indefinite problems (see Berestycki et al. [2]) and we should expect multiplicity
of solutions (see Ouyang [11]). In fact, by analyzing the Nehari set associated to
Φλ,µ:

Nλ,µ = {(u, v) ∈ X : Φ′λ,µ(u, v)(u, v) = 0},
one is lead to the conclusion that for some parameters, the Nehari set Nλ,µ is in fact

a manifold which split in two disjoint sets N+
λ,µ,N

−
λ,µ satisfying N+

λ,µ ∩ N−λ,µ = ∅
and N+

λ,µ ∩N−λ,µ = ∅, where

N+
λ,µ = {(u, v) ∈ X : Φ′′λ,µ(u, v)(u, v)2 > 0},
N−λ,µ = {(u, v) ∈ X : Φ′′λ,µ(u, v)(u, v)2 < 0}.

This suggest multiplicity of solutions, more precisely, the existence of critical points
to Φλ,µ in each manifold N+

λ,µ and N−λ,µ. In fact, let (λ1, φ1) be the first eigenpair of
−∆ in Ω with Dirichlet boundary conditions. We consider the following hypothesis
on f and g:

(H1) f, g ∈ L∞(Ω) and g(x) > 0 a.e. x ∈ Ω, f(x) ≥ 0 a.e. x ∈ Ω and f 6= 0.

Our main result reads as follows.

Theorem 1.1. Assume (H1) and that µ < λ1. Then there exists 0 < λ1(µ) <
λ∗(µ) <∞ such that

(1) For each λ ∈ (−∞, λ∗(µ)] problem (1.1) has at least one positive solution
(ūλ,µ, v̄λ,µ) ∈ N−λ,µ.

(2) For each λ ∈ (λ1(µ), λ∗(µ)] problem (1.1) has at least one positive solution
(uλ,µ, vλ,µ) ∈ N+

λ,µ.

By a positive solution we mean that both coordinates are positive functions. The
parameters λ1(µ), λ∗(µ) which appears in Theorem 1.1 are the so-called extremal
parameters (see Il’yasov [9]) and they describe the topological changes on the Nehari
set with respect to λ, µ. In fact, if λ < λ1(µ) we have that N+

λ,µ = ∅, while if

λ ≥ λ∗(µ), then Nλ,µ is no longer a C1 manifold. They can be found through the
study of the so-called nonlinear Rayleigh quotient

Rµ(u, v) :=

∫
(|∇u|2 + |∇v|2)− µ

∫
(|u|2 + |v|2)∫

uv
+

∫
g|v|q −

∫
f |u|p∫

uv
.

Since Nλ,µ is no longer a manifold when λ ≥ λ∗(µ), the technique used to prove
Theorem 1.1 can not be used to prove existence of solutions in this case, therefore,
we need a finer analysis over the Nehari sets. In this work we deal only with the
case where Nλ,µ is a manifold (and its limiting case), although some recent works
of Il’yasov and Silva [10], Silva and Macedo [13] suggests multiplicity of solutions
for λ > λ∗(µ).

This article is organized as follows: in Section 2 we study the fiber maps associ-
ated to Φλ,µ and the extremal parameters. In Section 3 analyze some topological
properties of the energy functional. In Section 4 we show existence of two positive
solutions to equation (1.1).



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 3

2. Non-linear generalized Rayleigh quotient and extremal
parameters

In this Section we establish some notation and technical results which will be used
throughout the paper. In particular we study the Nehari set and its decomposition
and we analyze the values of the parameters λ, µ for which Nλ,µ is a manifold.

From now on we denote w := (u, v) ∈ X . We equip X with the norm

‖w‖ =
(∫
|∇u|2 +

∫
|∇v|2

)1/2

.

If w ∈ X , we denote
‖w‖22 = ‖u‖22 + ‖v‖22.

For (λ, µ) ∈ R2, we recall the definition of the Nehari set

Nλ,µ = {w ∈ X \ {0} : Φ′λ,µ(w)w = 0}.
Note that the Nehari set can be written as

Nλ,µ = N+
λ,µ ∪N

0
λ,µ ∪N−λ,µ,

where

N+
λ,µ = {w ∈ X \ {0} : Φ′λ,µ(w)w = 0, Φ′′λ,µ(w)w2 > 0},
N 0
λ,µ = {w ∈ X \ {0} : Φ′λ,µ(w)w = 0, Φ′′λ,µ(w)w2 = 0},
N−λ,µ = {w ∈ X \ {0} : Φ′λ,µ(w)w = 0, Φ′′λ,µ(w)w2 < 0}.

Lemma 2.1. If N+
λ,µ,N

−
λ,µ are nonempty sets then N+

λ,µ,N
−
λ,µ are C1 manifolds of

codimension 1 in X . Moreover, w ∈ N+
λ,µ∪N

−
λ,µ is a critical point of (Φλ,µ)|N+

λ,µ∪N
−
λ,µ

if and only if w is a critical of Φλ,µ.

Since all critical points of Φλ,µ belongs to Nλ,µ, in order to find critical points to
Φλ,µ in X , we restrict our attention to critical points of Φλ,µ over Nλ,µ, however,
to apply Lemma 2.1 we need to understand the Nehari sets N+

λ,µ ∪N
−
λ,µ and N 0

λ,µ.

In fact, when N+
λ,µ∪N

−
λ,µ 6= ∅ and N 0

λ,µ = ∅ it is easy to show existence of solutions

to problem (1.1), however, when N 0
λ,µ we have to provide a more finer analysis over

the Nehari sets.
For λ, µ ∈ R and w ∈ X we introduce

Hλ,µ(w) = ‖w‖2 − µ‖w‖22 − 2λ

∫
uv.

First, let us characterize the Nehari set by using the Fibering Method of Po-
hozaev (see [12]): for each w ∈ X \ {0}, define ψλ,µ,w : [0,∞) → R by ψλ,µ,w(t) =
Φλ,µ(tw).

Proposition 2.2. For each λ, µ ∈ R and w ∈ X \ {0}, the function ψλ,µ,w is of
class C∞ on (0,∞). Moreover, the only three cases where ψλ,µ,w has a critical
point are:
Case 1: Hλ,µ(w) > 0.

(i) There is only one critical point at t−λ (w) ∈ (0,∞), and this point satisfies

ψ′′λ,w(t−λ,µ(w)) < 0 if only if
∫
f |u|p > 0;

Case 2: Hλ,µ(w) = 0.

(ii) ψλ,µ,w is constant equal to zero if and only if
∫
f |u|p,

∫
g|v|q = 0;



4 K. SILVA, S. M. SOUSA EJDE-2020/10

(iii) There is only one critical point at t−λ (w) ∈ (0,∞) and this point satisfies

ψ′′λ,w(t−λ,µ(w)) < 0 if only if
∫
f |u|p,

∫
g|v|q > 0;

Case 3: Hλ,µ(w) < 0.

(I) if
∫
f |u|p = 0 and

∫
g|v|q > 0 then there is only one critical point at

t+λ,µ(w) ∈ (0,∞) which satisfies ψ′′λ,µ,w(t−λ,µ(w)) > 0;

If
∫
f |u|p > 0 and

∫
g|v|q > 0 there are two possibilities:

(II) There are only two critical points for ψλ,w. One critical point at t−λ,µ(w)

with ψ′′λ,µ,w(t−λ,µ(w)) < 0 and the other one at t+λ,µ(w) with ψ′′λ,µ,w(t+λ,µ(w)) >

0. Moreover ψλ,µ,w is decreasing over the intervals [0, t+λ,µ(w)], [t−λ (w),∞]

and increasing over the interval [t+λµ(w), t−λ,µ(w)];

(III) The function ψλ,µ,w has only one critical point which is an inflection point
at t0λ,µ(w). Moreover, ψλ,µ,w is decreasing;

We start with the study of N+
λ,µ. Observe from Proposition 2.2 that if N+

λ,µ 6= ∅
then there exist (λ, µ) ∈ R2 and w ∈ X such that Hλ(w) < 0 or equivalently∫

|∇u|2 +
∫
|∇v|2 − µ

( ∫
|u|2 +

∫
|v|2
)

2
∫
uv

< λ,

therefore we are led to the study of the function

λmin(µ;w) :=
‖w‖2 − µ‖w‖22

2
∫
uv

, w ∈ X ,
∫
uv > 0. (2.1)

Now we turn our attention to the Nehari set N 0
λ,µ. From Proposition 2.2 we have

that if w ∈ N 0
λ,µ, then ∫

f |u|p > 0, and

∫
uv > 0.

Let us introduce the set (the open subset of X )

X+ :=
{
w ∈ X :

∫
f |u|p > 0,

∫
uv > 0

}
,

so N 0
λ,µ ⊂ X+. For each w ∈ X+, consider the corresponding so-called scalar fibered

Rayleigh quotient (see Il’yasov [9])

Rµ(tw) =

∫
(|∇u|2 + |∇v|2) dx− µ

∫
(|u|2 + |v|2) dx∫

uv dx

+
tq−2

∫
g|v|q dx− tp−2

∫
f |u|p dx∫

uv dx
.

As was shown in [9] we have that

w ∈ N 0
λ,µ, if and only if Rµ(w) = λ,

d

dt
R(tw)|t=1 = 0.

and hence the extremal values of [0,∞) 3 t 7→ Rµ(tw) provide regions of parameters
where N 0

λ,µ = ∅. Assume that µ < λ1 and observe that [0,∞) 3 t 7→ Rµ(tw) has
two extremal values. The first one is a local minimum attained at t = 0, indeed, it
corresponds to λmin(µ;w) as defined in (2.1): one can easily see that

λmin(µ;w) ≥ 1− µ

λ1
> 0, ∀µ < λ1, ∀w ∈ X+.



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 5

The second one corresponds to a local maximum which can be computed by using
standard calculus in the following way:

d

dt
Rµ(tw) =

(q − 2)tq−3
∫
g|v|qdx− (p− 2)tp−3

∫
f |u|pdx∫

uv dx
= 0, t > 0,

if and only if

(q − 2)

∫
g|v|qdx− (p− 2)tp−q

∫
f |u|γdx = 0,

and hence

t0(w) =
( (q − 2)

∫
g|v|qdx

(p− 2)
∫
f |u|pdx

) 1
p−q

(2.2)

is the critical point of [0,∞) 3 t 7→ Rµ(tw) which corresponds to a global maximum.
Therefore we have the nonlinear generalized Rayleigh quotient

λmax(µ;w) := max
t≥0

Rµ(tw) =
1

2
∫
uv

(
‖w‖2 − µ‖w‖22 + Cp,q

( ∫
g|v|qdx

) p−2
p−q( ∫

f |u|pdx
) q−2
p−q

)
,

where Cp,q > 0 is given by

Cp,q =
(q − 2

p− 2

) q−2
p−q −

(q − 2

p− 2

) p−2
p−q

.

Remark 2.3. We observe here that to study the scalar fibered Rayleigh quotient,
there is no need to assume that w ∈ X+; however, [0,∞) 3 t 7→ Rµ(tw) has a global
maximum if, and only if w ∈ X+. Furthermore, note that if µ < λ1 and

∫
uv >0,

then λmin(µ;w) is just the local minimum of [0,∞) 3 t 7→ Rµ(tw) which is attained
at t = 0.

The functions λmin(µ;w) and λmax(µ;w) have the following geometrical inter-
pretation, with respect to the fiber maps, which can be proved from Proposition
2.2 and their definitions.

Proposition 2.4. The following holds:

(1) For each µ < λ1 and λ ∈ R we have that N−λ,µ 6= ∅. Moreover N+
λ,µ 6= ∅ if,

and only if λ > λmin(µ,w) and µ < λ1.
(2) For each µ < λ1 and w ∈ X+ we have that: λmax(µ;w) is the unique

parameter λ > 0 for which the fiber map ψλ,w has a critical point with
second derivative zero at t(w). If λmin(µ,w) < λ < λmax(µ;w), then ψλ,µ,w
satisfies II) of the Proposition 2.2 while if λ > λmax(µ;w), then ψλ,w is
decreasing and has no critical points.

Let us consider the following critical values:

λ∗1(µ) = inf
{
λmin(µ;w) : w ∈ X :

∫
uv dx > 0

}
, (2.3)

λ∗(µ) = inf
{
λmax(µ;w) : w ∈ X+

}
. (2.4)

Lemma 2.5. For each µ < λ1 it holds 0 < λ∗1(µ) < λ∗(µ) < +∞. Moreover,

(i) λ∗1(µ) = λ1 − µ.
(ii) There exists a minimizer w∗ := (u∗, v∗) ∈ X+ of (2.4), which means

λmax(µ;w∗) = λ∗(µ).



6 K. SILVA, S. M. SOUSA EJDE-2020/10

Proof. (i) Indeed, we have

λmin(µ;w) ≥
(

1− µ

λ1

) ‖w‖2
2
∫
uv
, (2.5)

and ∫
uv ≤ ‖u‖2‖v‖2 ≤

( 1√
λ1

‖u‖
)( 1√

λ1

‖v‖
)

=
1

λ1
‖u‖‖v‖ ≤ 1

λ1

(∫ |∇u|2 + |∇v|2

2

)
=

1

λ1

‖w‖2

2
.

Then we obtain

λ1 ≤ inf
‖(u, v)‖2

2
∫
uv

. (2.6)

It follows from (2.6) and (2.5) that

λ∗1(µ) ≥ λ1 − µ.

Since
∫
φ2

1 > 0 and λmin(µ, φ1, φ1) = λ1 − µ it follows that λ∗1(µ) = λ1 − µ.
(ii) Now let us prove there exists w∗ ∈ X+ such that λ∗(µ) = λmax(µ;w∗).

Choose a sequence wn := (un, vn) ∈ X+ such that λmax(µ,wn)→ λ∗(µ) as n→∞
and since λmax(µ, tw) = λmax(µ;w) for t > 0, we can assume without loss of
generality that ‖wn‖ = 1 and therefore wn ⇀ w in X and wn → w in Lp(Ω)×Lq(Ω).
Note that

∫
uv > 0 because

λmax(µ;wn) ≥
(

1− µ

λ1

) ‖wn‖2
2
∫
unvn

=
(

1− µ

λ1

) 1

2
∫
unvn

, ∀n,

and on the contrary, we would have λmax(µ;wn)→ +∞ which is clearly a contra-
diction. It follows that u, v 6= 0. We claim that

∫
f |u|p > 0, indeed suppose on the

contrary that
∫
f |u|p = 0. Since

λ∗max(µ;wn) ≥ Cp,q
( ∫

g|vn|qdx
) p−2
p−q( ∫

f |un|pdx
) q−2
p−q

, ∀n, (2.7)

and
∫
g|v|q > 0 we conclude that λmax(µ;wm) → +∞ which is a contradiction,

therefore
∫
f |u|p > 0. We denote by ū = u

‖w‖ , v̄ = v
‖w‖ , then w̄ = (ū, v̄) satisfies

‖w̄‖ = 1,
∫
ūv̄ > 0 and

∫
f |ū|p,

∫
g|v̄|q > 0. We claim that wn → w in X , indeed if

not, by the weak lower semi-continuity of the norm, we have

λmax(µ; w̄) < lim inf
n→∞

λmax(wn) = λ∗(µ) (2.8)

which is an absurd and hence λ∗(µ) = λmax(µ; w̄). By defining w∗ = w̄ the proof
is complete. �

Proposition 2.6. Let µ < λ1, then N 0
λ∗(µ),µ 6= ∅. Moreover, each w ∈ N 0

λ∗(µ),µ

satisfies

2
(
−∆u− µu− λ∗(µ)v

)
− pf(x)|u|p−2u = 0,

2
(
−∆v − µv − λ∗(µ)u

)
+ qg(x)|v|q−2v = 0,

(2.9)

Proof. From Lemma 2.5 there exists w ∈ X \ {0} such that λmax(µ;w) = λ∗(µ).
From the definition of λmax(µ;w) it follows that N 0

λ∗(µ),µ 6= ∅. To prove that each



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 7

w ∈ N 0
λ∗(µ),µ satisfies (2.9), we observe that λ′max(µ;w)w̄ = 0 for all w̄ = (ū, v̄) ∈ X ,

hence we obtain

0 =2
(∫
∇u∇ū− µ

∫
uū− λ∗(µ)

∫
vū
)

− p
(q − 2

p− 2

) p−2
p−q
( ∫ g|v|q∫

f |u|p
) p−2
q−2

∫
f |u|p−2uū,

0 =2
(∫
∇v∇v̄ − µ

∫
vv̄ − λ∗(µ)

∫
uv̄
)

− q
(q − 2

p− 2

) p−2
p−q
( ∫ g|v|q∫

f |u|p
) p−2
q−2

∫
g|v|q−2vv̄.

(2.10)

For w ∈ N 0
λ∗(µ),µ we have(q − 2

p− 2

) p−2
p−q
( ∫ g|v|q∫

f |u|p
) p−2
q−2

= 1 . (2.11)

Then, from (2.10) and (2.11) we conclude the proof. �

Corollary 2.7. Let µ < λ1. Then

(i) For each λ ∈ R we have that N−λ,µ 6= ∅.
(ii) N+

λ,µ 6= ∅ if, and only if λ > λ∗1(µ).

(iii) N 0
λ,µ 6= ∅ if, and only if λ ≥ λ∗(µ).

Proof. (i) Given λ ∈ R there exists wn := (un, vn) ∈ X+ such that ‖un‖ = 1, vn 6= 0
and vn → 0 in H1

0 (Ω), so

lim
n→∞

Hλ,µ(wn) ≥ lim
n→∞

(
1− µ

λ1

)
+ ‖vn‖2 − µ‖vn‖22 − 2λ

∫
unvn

=
(

1− µ

λ1

)
> 0,

then there exists n0 ∈ N such that for all n ≥ n0 we obtain Hλ,µ(wn) > 0 and from
Proposition 2.2 we conclude that N−λ,µ 6= ∅.

(ii) Suppose N+
λ,µ 6= ∅ and take w ∈ N+

λ,µ. By Proposition 2.2 we conclude that

Hλ,µ(w) < 0 which implies
‖w‖2−µ‖w‖22

2
∫
uv

< λ, therefore λ∗1(µ) < λ.

Now suppose that λ∗1(µ) < λ and take w = (φ1, φ1). It follows that

λ∗1(µ) =
‖w‖2 − µ‖w‖22

2
∫
φ2

1

< λ,

hence Hλ,µ(w) < 0. Since
∫
g|φ1|q > 0, from Proposition 2.2 we conclude that

t+λ,µ(w)w ∈ N+
λ,µ.

(iii) Suppose N 0
λ,µ 6= ∅. We know that w ∈ N 0

λ,µ if, and only if

Rµ(w) = λ,
d

dt
R(tw)|t=1 = 0,

and therefore by definition of λ∗(µ), we conclude that λ∗(µ) ≤ λ.
Now observe from Lemma 2.5 that there exists w∗ ∈ X+ such that w∗ ∈ N 0

λ∗(µ),µ.

Moreover there exists wn := (un, vn) ∈ X+ such that ‖un‖ = 1, vn 6= 0 and vn → 0



8 K. SILVA, S. M. SOUSA EJDE-2020/10

in H1
0 (Ω), then

lim
n→∞

λmax(µ;wn) ≥ lim
n→∞

1

2
∫
unvn

((
1− µ

λ1

)
+ ‖vn‖2 − µ‖vn‖22

)
=∞,

therefore, from the continuity of λmax(µ;w) with respect to w, given λ ≥ λ∗(µ)
we there exists w ∈ X+ such that λmax(µ;w) = λ and from Proposition 2.4 we
conclude that N 0

λ,µ 6= ∅. �

3. Topological properties of the energy functional

In this Section we study the energy functional Φλ,µ, in particular, we show that
Φλ,µ has some well know topological properties when restricted to the Nehari set,
as for example coerciveness, which allow us to minimize over the Nehari manifolds
N−λ,µ and N+

λ,µ.
For λ > 0 we define

N̂−λ,µ =
{
w ∈ N−λ,µ : Hλ,µ(w) ≤ 0

}
.

Proposition 3.1. For each µ < λ1 and λ ∈ R, we have the following:

(i) There exists a constant C > 0 such that ‖w‖ ≤ C for all w ∈ N+
λ,µ ∪ N̂

−
λ,µ.

(ii) The functional Φλ,µ restricted to N+
λ,µ∪N

−
λ,µ is coercive that is if wn ∈ N−λ,µ

is such that ‖wn‖ → ∞ as n→∞, then Φλ,µ(wn)→∞ as n→∞.

Proof. Assume that wn = (un, vn) ∈ N+
λ,µ ∪ N

−
λ,µ satisfies ‖wn‖ → ∞. We claim

that ∫ ∣∣ un
‖wn‖

∣∣p → 0, as n→∞. (3.1)

If not, then there exists C̄ > 0 such that, up to a subsequence,
∫ ∣∣∣ un
‖wn‖

∣∣∣p > C̄.

Denote by ūn = un
‖wn‖ and v̄n = vn

‖wn‖ . Since wn ∈ N+
λ,µ ∪N

−
λ,µ, we have

0 = 1− µ
(∫
|ūn|2 + |v̄n|2

)
− 2λ

∫
ūnv̄n + ‖wn‖q−2

∫
g|v̄n|q − ‖wn‖p−2

∫
f |ūn|p.

(3.2)
By Sobolev embedding and Poincare’s inequality there exist constants C1, C2, C3 >
0 such that

∫
g|v̄n|q ≤ C1,

∫ (
|ūn|2 + |v̄n|2

)
≤ C2 and

∫
ūnv̄n ≤ C3. It follows from

(3.2) that

0 = 1− µ
(∫
|ūn|2 + |v̄n|2

)
− 2λ

∫
ūnv̄n + ‖wn‖q−2

∫
g|v̄n|q

− ‖wn‖p−2

∫
f |ūn|p

≤ 1 + |µ|C2 + 2|λ|C3 + C1‖wn‖q−2 − C̄‖wn‖p−2,∀n,

which is a contradiction since p > q and therefore (3.1) is true.
Let us prove (i). Suppose on the contrary that there exists a sequence wn ∈

N+
λ,µ∪N̂

−
λ,µ such that ‖wn‖ → ∞ as n→∞. From (3.1) we obtain that

∫
|ūn|2 → 0

and since Hλ,µ(wn) ≤ 0 and µ < λ1 we conclude that

0 ≥ 1− µ
(∫
|ūn|2 + |v̄n|2

)
− 2λ

∫
ūnv̄n

≥
(

1− µ

λ1

)
− 2λ

∫
ūnv̄n



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 9

→
(

1− µ

λ1

)
> 0, n→∞,

which is a contradiction and therefore there exists a constant C > 0 such that
‖w‖ ≤ C for all w ∈ N+

λ,µ ∪ N̂
−
λ,µ.

Let us prove (ii): Assume that ‖wn‖ → ∞ as n → ∞. From (3.1) and (3.2) we
conclude that

‖wn‖q−2

∫
g|v̄n|q − ‖wn‖p−2

∫
f |ūn|p − µ

∫
|v̄|2 = o(1)− 1. (3.3)

Now observe that

Φλ,µ(wn) = ‖wn‖2
(1

2
− µ1

2

∫
|ūn|2 + |v̄n|2 − λ

∫
ūnv̄n

)
+ ‖wn‖2

(‖wn‖q−2

q

∫
g|v̄n|q −

‖wn‖p−2

p

∫
f |ūn|p

)
.

(3.4)

Observe that Φλ,µ(w) > 0 for all w ∈ N−λ,µ. If we assume on the contrary that

Φλ,µ(wn) does not converge to ∞ then from (3.4) we are forced to assume that

‖wn‖q−2

q

∫
g|v̄n|q −

‖wn‖p−2

p

∫
f |ūn|p −

1

2
µ

∫
|v̄|2 = o(1)− 1

2
. (3.5)

from (3.3) and (3.5) we obtain

‖wn‖p−2

∫
f |ūn|p = o(1) +

2

p

(q − 2

q − p
)(

1− µ
∫
|v̄n|2

)
, (3.6)

‖wn‖q−2

∫
g|v̄n|q = o(1) +

2

p

(q − 2

q − p
)(

1− µ
∫
|v̄n|2

)
. (3.7)

Once µ < λ1 and q < p it follows from (3.6), (3.7) that

‖wn‖p−2

∫
f |ūn|p ≤ o(1) +

2

p

(q − 2

q − p
)(

1− µ

λ1

)
,

‖wn‖q−2

∫
g|v̄n|q ≤ o(1) +

2

p

(q − 2

q − p
)(

1− µ

λ1

)
,

which is a contradiction and therefore Φλ,µ(wn)→∞ as n→∞. �

From Proposition 3.1 we have the following result.

Corollary 3.2. Suppose that µ < λ1 and λ ∈ R. Then there exists a constant
C > 0 such that Φλ(w) ≥ −C, for all w ∈ N+

λ,µ ∪N
−
λ,µ.

Lemma 3.3. For each µ < λ1 and λ ∈ R there exists a constant C > 0 such that
‖w‖ ≥ C, for all w ∈ N−λ,µ. Moreover, if A ⊂ N−λ,µ is a bounded set, then ‖u‖p ≥ C
for each (u, v) ∈ A.

Proof. Indeed, suppose on the contrary that there exists wn = (un, vn) ∈ N−λ,µ such

that ‖wn‖ → 0. If vn = 0 for all n the proof is immediate, therefore there is no loss of
generality in assuming that vn 6= 0 for all n. Moreover from Proposition 2.2 we also
have that un 6= 0 for all n. Define ūn = un

‖wn‖ and v̄n = vn
‖wn‖ and w̄n = (ūn, v̄n).

It follows that w̄n ⇀ (u0, v0) in X and w̄n → (u0, v0) in Lp(Ω) × Lq(Ω). Once
wn ∈ N−λ,µ we know that

1− µ‖w̄n‖22 − 2λ

∫
ūnv̄n = ‖wn‖p−2

∫
f |ūn|p − ‖wn‖q−2

∫
g|v̄n|q, ∀n, (3.8)



10 K. SILVA, S. M. SOUSA EJDE-2020/10

and

1− ‖w̄n‖22 − 2λ

∫
ūnv̄n + (q − 1)‖wn‖q−2

∫
g|v̄n|q − (p− 1)‖wn‖p−2

∫
f |ūn|p < 0,

and hence

(q − 2)‖wn‖q−2

∫
g|v̄n|q − (p− 2)‖wn‖p−2

∫
f |ūn|p < 0, ∀n,

which implies
1

‖wn‖p−q
<
p− 2

q − 2

∫
f |ūn|p∫
g|v̄n|q

, ∀n,

Hence
∫
g|v̄n|q → 0 as n→∞ which combined with (3.8) gives us an absurd since

µ < λ1 and therefore N−λ is bounded always from the origin.

Now assume that A ⊂ N−λ,µ is a bounded set. For each w ∈ A we have that

‖w‖2 − µ‖w‖22 − 2λ

∫
uv +

∫
g|v|q −

∫
f |u|p = 0. (3.9)

If on the contrary we can find wn ∈ A such that un → 0 in Lp(Ω), then since A is
bounded, from (3.9) we obtain ‖wn‖2 − µ‖vn‖22 +

∫
g|vn|q = o(1) and once µ < λ1

we conclude that ‖wn‖ = o(1) that is a contradiction and therefore, there exists
C > 0 such that ‖u‖p ≥ C for each w ∈ A. �

4. Existence of solutions in (−∞, λ∗(µ)]

In this section, by using the properties of the fiber maps, we prove existence of
positive solutions to the problem (1.1) for λ ∈ (−∞, λ∗(µ)] and µ < λ1.

Remark 4.1. We claim that there is no non-negative solution of (1.1) for µ > λ1

and λ > 0. Indeed, take φ1 ∈ H1
0 (Ω) and let w := (u, v) ∈ X be a non-negative

solution for (1.1), then∫
∇u∇φ1 = λ1

∫
uφ1 = µ

∫
uφ1 + λ

∫
vφ1 +

∫
f |u|p−2uφ1 ≥ µ

∫
uφ1

we obtain

(λ1 − µ)

∫
uφ1 ≥ 0

which implies that u = v = 0, since µ > λ1. Therefore there is no non-negative
solution of (1.1) for µ > λ1 and λ > 0. If w is a positive solution, then the same
argument holds for all µ ≥ λ1 and λ > 0.

For λ ∈ R define

M̂λ,µ := {w ∈ X : ψλ,µ,w satisfies (I) or (II) of Proposition 2.2},
and

M̂−λµ :=
{
w ∈ X \ {0} : Hλ,µ(w) ≥ 0,

∫
f |u|p > 0

}
.

For λ ∈ R, let J−λ,µ : M̂λ,µ ∪ M̂−λ,µ → R and J+
λ,µ : M̂λ,µ → R be defined by

J−λ,µ(w) = Φλ,µ(t−λ,µ(w)w), and J+
λ,µ(w) = Φλ,µ(t+λ,µ(w)w).

Remark 4.2. Observe from Proposition 2.2 that N+
λ,µ ∪N

−
λ,µ ⊂ M̂λ,µ ∪M̂−λ,µ and

from Corollary 2.7 we have that N+
λ,µ 6= ∅ if λ > λ1(µ) and N−λ,µ 6= ∅ if λ ∈ R.

Moreover J−λ,µ, J
+
λ,µ are the restrictions of Φλ,µ to N−λ,µ and N+

λ,µ respectively.



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 11

We consider the following constrained minimization problems

Ĵ−λ,µ := inf{J−λ,µ(w) : w ∈ N−λ,µ}, ∀λ ∈ R,

and

Ĵ+
λ,µ := inf{J+

λ,µ(w) : w ∈ N+
λ,µ}, ∀λ > λ1(µ).

Proposition 4.3. It holds:

• For each λ ∈ (λ1(µ), λ∗(µ)) there there exists wλ := (uλ, vλ) ∈ N+
λ,µ such

that Ĵ+
λ,µ = J+

λ,µ(wλ).

• For each λ ∈ (−∞, λ∗(µ)) there there exists w̄λ := (ūλ, v̄λ) ∈ N−λ,µ such

that Ĵ−λ,µ = J−λ,µ(w̄λ).

Proof. Firstly, we start with Ĵ+
λ,µ. We may suppose that there exists wn :=

(un, vn) ∈ N+
λ,µ such that J+

λ,µ(wn) → Ĵ+
λ,µ. From Proposition 3.1 we have wn ⇀

w := (u, v) in X and wn → w in Lp(Ω)× Lq(Ω). Since

Φλ,µ(w) ≤ lim inf Φλ,µ(wn) = Ĵ+
λ,µ, (4.1)

and by Proposition 2.2 we have that Ĵ+
λ,µ < 0, we conclude that w 6= 0. We claim

that wn → w in X . Indeed suppose on the contrary that it is false. By one hand
note from Proposition 2.2 that Hλ,µ(w) < lim infn→∞Hλ,µ(wn) ≤ 0 and since

λ ∈ (λ1, λ
∗(µ)) we conclude that w ∈ M̂λ,µ. On the other hand

0 = ψ′λ,µ,w(t+λ,µ(w)) < lim inf
n→∞

ψ′λ,µ,wn(t+λ,µ(w)),

and hence t+λ,µ(w) > 1 which implies that

J+
λ,µ(w) < lim inf

n→∞
Φλ,µ(w) < lim inf

n→∞
Φλ,µ(wn) = Ĵ+

λ,µ,

which is a contradiction. Therefore wn → w in X , w ∈ N+
λ,µ and Ĵ+

λ,µ = J+
λ,µ(w).

Now we consider wn := (un, vn) ∈ N−λ,µ such that J−λ,µ(wn) → Ĵ−λ,µ. From

Proposition 3.1 we have wn ⇀ w := (u, v) in X and wn → w in Lp(Ω) × Lq(Ω).
Then from Lemma 3.3 we have that u 6= 0 and hence from Proposition 2.2, t−λ,µ(w)
is well defined. We claim that wn → w in X , so suppose that is not true. Observe
that

0 = ψ′λ,µ,w(t−λ,µ(w)) < lim inf
n→∞

ψ′λ,µ,wn(t−λ,µ(w)),

and hence t+λ,µ(wn) < t−λ,µ(w) < 1 for sufficiently large n in case t+λ,µ(wn) is well

defined and tλ,µ(w) < 1 in case t+λ,µ(wn) is not defined. In both cases we have

J−λ,µ(w) < lim inf
n→∞

Φλ,µ(t−λ,µ(w)wn) ≤ lim inf
n→∞

Φλ,µ(wn) = Ĵ−λ,µ,

that is an absurd. Therefore wn → w in X , w ∈ N−λ,µ and Ĵ−λ,µ = J−λ,µ(w). �

The next Proposition will be useful in order to prove existence of solutions when
λ ≥ λ∗(µ).

Proposition 4.4. Fix µ < λ1 and take w ∈ X \ 0 such that
∫
uv > 0. Let I ⊂ R

be an open interval such that t∓λ,µ(w) are well defined for all λ ∈ I. It holds:

(i) The functions I 3 λ 7→ t∓λ,µ(w) are C1. Moreover, I 3 λ 7→ t−λ,µ(w) is

decreasing while I 3 λ 7→ t+λ,µ(w) is increasing.



12 K. SILVA, S. M. SOUSA EJDE-2020/10

(ii) The functions I 3 λ 7→ J∓λ,µ(w) are continuous and decreasing.

Proof. (i) For each w ∈ X \ 0 fixed we define

F (λ, t) = Hλ,µ(tu, tv) +G(v)− F (u).

Since t∓λ,µ(w)w ∈ N∓λ,µ, it follows that

F (λ, t∓λ,µ(w)w) = 0,

∂

∂t
F (λ, t∓λ,µ(w)) 6= 0,

which implies from the implicit function theorem that t∓λ,µ(w) is C1 and

∂

∂λ
t∓λ,µ(w) =

2
∫
uv

ψ′′λ,µ,w(t∓λ,µ(w))
,

therefore, ∂
∂λ t

+
λ,µ(w) > 0 and ∂

∂λ t
−
λ,µ(w) < 0.

(ii) Indeed,
∂

∂λ
J∓λ,µ(w) = −

∫
uv;

therefore, J∓λ,µ is decreasing. �

Proposition 4.5. For each µ < λ1, there exists w ∈ N+
λ∗(µ),µ and w̄ ∈ N−λ∗(µ),µ

such that Ĵ+
λ∗(µ),µ = J+

λ∗(µ),µ(w) and Ĵ−λ∗(µ),µ = J−λ∗(µ),µ(w̄).

Proof. Take λn ↑ λ∗(µ) and wn := (un, vn) ∈ N−λn,µ with Ĵ−λn,µ = Jλn,µ(wn). From
Lemma 2.1 we have

−∆un − µun − λ∗(µ)vn − f(x)|un|p−2un = 0,

−∆vn − µvn − λ∗(µ)un + g(x)|vn|q−2vn = 0,

for each n. Using similar arguments to those in Proposition 3.1 and Lemma 3.3,
we can show that there exist constants C, c > 0 such that c ≤ ‖wn‖ ≤ C. We
can suppose without loss generality that wn ⇀ w := (u, v) in X and wn → w in
Lp(Ω)× Lq(Ω). Hence wn → w 6= 0 in X and we conclude that

−∆u− µu− λ∗(µ)v − f(x)|u|p−2u = 0,

−∆v − µv − λ∗(µ)u+ g(x)|v|q−2v = 0,
(4.2)

We claim that w ∈ N−λ∗(µ),µ. If not, then w ∈ N 0
λ∗(µ),µ and from Proposition 2.6,

2 (−∆u− µu− λ∗(µ)v)− pf(x)|u|p−2u = 0,

2 (−∆v − µv − λ∗(µ)u) + q|v|q−2g(x)v = 0.
(4.3)

From (4.2) and (4.3) we have

(2− p)f(x)|u|p−2u = 0,

(2 + q)g(x)|v|q−2v = 0,
(4.4)

which implies w = 0, an absurd. Therefore w ∈ N−λ∗(µ),µ and hence J−λ∗(µ),µ(w) ≥
Ĵ−λ∗(µ),µ. To conclude the proof we need to show that J−λ∗(µ),µ(w) = Ĵ−λ∗(µ),µ so



EJDE-2020/10 MULTIPLICITY OF SOLUTIONS FOR GRADIENT TYPE SYSTEMS 13

suppose on the contrary that J−λ∗(µ),µ(w) > Ĵ−λ∗(µ),µ. Given ε > 0 there exists

z ∈ N−λ∗(µ),µ such that

0 < J−λ∗(µ),µ(z)− Ĵ−λ∗(µ),µ < ε. (4.5)

From Proposition 4.4 we can also find N > 0 such that

0 < J−λn,µ(z)− J−λ∗(µ),µ(z) < ε, ∀ n > N. (4.6)

From (4.5) and (4.6) we conclude that

Ĵ−λn,µ = J−λ∗(µ),µ(w) + o(1) > Ĵ−λ∗(µ),µ + o(1)

> J−λn,µ(z)− 2ε+ o(1) ≥ Ĵ−λn,µ − 2ε+ o(1),

which is a contradiction and hence J−λ∗(µ),µ(w) = Ĵ−λ∗(µ),µ. A similar proof can be

carried out for Ĵ+
λ∗ . �

Proof of Theorem 1.1. From Propositions 4.3 and 4.5, it follows that there exist
wλ,µ := (uλ,µ, vλ,µ) ∈ N+

λ,µ and w̄λ,µ := (ūλ,µ, v̄λ,µ) ∈ N−λ,µ, such that Ĵ+
λ,µ =

J+
λ,µ(wλ,µ) and Ĵ−λ,µ = J−λ,µ(w̄λ,µ). For simplicity we define w := wλ,µ and w̄ :=

w̄λ,µ, then from Lemma 2.1 we have that w and w̄ are solutions of problem (1.1).
Let us prove now that w and w̄ can be chosen as positive functions. We do it

only to w̄ since for w the calculations are similar. First, observe that Hλ,µ(|w̄|) ≤
Hλ,µ(w̄), where |w̄| := (|ū|, |v̄|). We claim that Hλ,µ(|w̄|) = Hλ,µ(w̄). Suppose on
the contrary that Hλ,µ(|w̄|) < Hλ,µ(w̄).

Case 1: λ ∈ (−∞, λ∗(µ)). From Proposition 2.2 and since |ū| 6= 0, there exist
t− := t−λ,µ(|w̄|) > 0 such that t−|w̄| ∈ N−λ,µ. Once Hλ,µ(|w̄|) < Hλ,µ(w̄), we have

0 = ψ′λ,µ,|w̄|(t
−) < ψ′λ,µ,w̄(t−),

which from Proposition 2.2 implies t− < 1 and in this case t+λ,µ(w̄) is defined; we

also have that t+λ,µ(w̄) < t− < 1. It follows that

Φλ,µ(t−|w̄|) =
(t−)2

2
Hλ,µ(|w̄|) +

(t−)q

q

∫
g|v̄|q − (t−)p

p

∫
f |ū|p

<
(t−)2

2
Hλ,µ(w̄) +

(t−)q

q

∫
g|v̄|q − (t−)p

p

∫
f |ū|p

= Φλ,µ(t−w̄) < Φλ,µ(w̄) = Ĵ−λ,µ

which is a contradiction and therefore Hλ,µ(|w̄|) = Hλ,µ(w̄).

Case 2: λ = λ∗(µ). Indeed, we claim that Ĵ−λ,µ = Φλ,µ(w̄) < 0 so Hλ,µ(w̄) < 0.

If not, then Hλ,µ(w̄) ≥ 0 and by Proposition 2.2 we obtain that Ĵ−λ,µ ≥ 0 which

is an absurd. By the definition of λ∗(µ) and Propositions 2.2 and 2.4, there exists
t := tλ,µ(|w̄λ,µ|) > 0 such that t|w̄| ∈ N−λ,µ ∪N 0

λ,µ and hence

0 = ψ′λ,µ,|w̄|(t) < ψ′λ,µ,w̄(t).

From Proposition 2.2 it follows that t < 1. Then

Φλ,µ(t|w̄|) < Φλ,µ(tw̄) < Φλ,µ(w̄) = Ĵ−λ,µ (4.7)



14 K. SILVA, S. M. SOUSA EJDE-2020/10

which is a contradiction. Therefore Hλ,µ(|w̄|) = Hλ,µ(w̄) which implies that

ψ′λ,µ,|w̄|(1) = ψ′λ,µ,w̄(1) = 0,

ψ′′λµ,|w̄|(1) = ψ′′λ,µ,w̄(1) < 0.
(4.8)

Therefore we can assume that w, w̄ ≥ 0. Moreover, one can easily see from (1.1) that
the functions u, v, ū, v̄ are non-zero. From standard regularity theory we conclude
that u, v, ū, v̄ ∈ C1,α(Ω) for some α ∈ (0, 1) and they are positive everywhere in
Ω. �

References

[1] C. O. Alves, D. C. de Morais Filho, M. A. S. Souto; On systems of elliptic equations involving
subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000), no. 5, Ser. A: Theory

Methods, 771–787. MR 1776922.

[2] Henri Berestycki, Italo Capuzzo-Dolcetta, Louis Nirenberg; Variational methods for indefinite
superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl.,

2 (1995), no. 4, 553–572. MR 1356874.

[3] Yuri Bozhkov, Enzo Mitidieri; Existence of multiple solutions for quasilinear systems via
fibering method, J. Differential Equations, 190 (2003), no. 1, 239–267. MR 1970963.

[4] K. J. Brown; Spatially inhomogeneous steady state solutions for systems of equations describ-

ing interacting populations, J. Math. Anal. Appl., 95 (1983), no. 1, 251–264. MR 710432.
[5] Ph. Clément, D. G. de Figueiredo, E. Mitidieri; Positive solutions of semilinear elliptic

systems, Comm. Partial Differential Equations 17 (1992), no. 5-6, 923–940. MR 1177298.

[6] D. G. Costa, C. A. Magalhães; A variational approach to subquadratic perturbations of elliptic
systems, J. Differential Equations, 111 (1994), no. 1, 103–122. MR 1280617.

[7] Edcarlos D. da Silva; Multiplicity of solutions for gradient systems using Landesman-Lazer
conditions, Abstr. Appl. Anal., (2010), Art. ID 237826, 22. MR 2720025.

[8] Djairo G. deFigueiredo; Nonlinear elliptic systems, An. Acad. Brasil. Ciênc. 72 (2000), no. 4,

453–469. MR 1800060.
[9] Yavdat Ilyasov; On extreme values of Nehari manifold method via nonlinear Rayleigh’s quo-

tient, Topol. Methods Nonlinear Anal., 49 (2017), no. 2, 683–714. MR 3670482.

[10] Yavdat Ilyasov, Kaye Silva; On branches of positive solutions for p-Laplacian problems at
the extreme value of the Nehari manifold method, Proc. Amer. Math. Soc., 146 (2018), no. 7,

2925–2935. MR 3787354.

[11] Tiancheng Ouyang; On the positive solutions of semilinear equations ∆u+ λu+ hup = 0 on
compact manifolds. II, Indiana Univ. Math. J., 40 (1991), no. 3, 1083–1141. MR 1129343.

[12] S. I. Pokhozhaev; The fibration method for solving nonlinear boundary value problems, Trudy

Mat. Inst. Steklov., 192 (1990), 146–163, Translated in Proc. Steklov Inst. Math., 1 992, no.
3, 157–173, Differential equations and function spaces (Russian). MR 1097896.

[13] Kaye Silva, Abiel Macedo; Local minimizers over the Nehari manifold for a class of concave-
convex problems with sign changing nonlinearity, J. Differential Equations, 265 (2018), no. 5,
1894–1921. MR 3800105.

[14] Wenming Zou; Multiple solutions for asymptotically linear elliptic systems, J. Math. Anal.
Appl., 255 (2001), no. 1, 213–229. MR 1813819.

Kaye Silva

Instituto de Matemática e Estat́ıstica, Universidade Federal de Goiás, 74001-970,
Goiânia, GO, Brazil

Email address: kayesilva@ufg.br

Steffânio Moreno Sousa
Instituto de Matemática e Estat́ıstica Universidade Federal de Goiás, 74001-970, Goiânia,

GO, Brazil

Email address: steffaniom@gmail.com


	1. Introduction
	2. Non-linear generalized Rayleigh quotient and extremal parameters
	3. Topological properties of the energy functional
	4. Existence of solutions in (-,*()]
	References