Tenth MSU Conference on Differential Equations and Computational Simulations.
Electronic Journal of Differential Equations, Conference 23 (2016), pp. 187–194.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE SOLUTIONS FOR 3 × 3 ELLIPTIC BI-VARIATE
INFINITE SEMIPOSITONE SYSTEMS WITH COMBINED
NONLINEAR EFFECTS
JINGLONG YE, JAFFAR ALI
Abstract. We study the existence of positive solutions to 3 × 3 bi-variate
systems of reaction diffusion equations with Dirichlet boundary conditions. In
particular, we consider systems where the reaction terms approach −∞ near
the origin and satisfy some combined sublinear conditions at ∞. We use the
method of sub-super solutions to establish our results.
1. Introduction
We study nonlinear elliptic 3 × 3 bi-variate systems of the form
g1 (u2 , u3 )
in Ω,
−∆u1 = λ
1
uα
1
g2 (u3 , u1 )
in Ω,
−∆u2 = λ
2
uα
2
g3 (u1 , u2 )
−∆u3 = λ
in Ω,
3
uα
3
u1 = u2 = u3 = 0; on ∂Ω
and
g1 (u2 , u3 )
in Ω,
uα
2
g2 (u3 , u1 )
−∆u2 = λ
in Ω,
uα
3
g3 (u1 , u2 )
−∆u3 = λ
in Ω,
uα
1
u1 = u2 = u3 = 0; on ∂Ω,
(1.1)
−∆u1 = λ
(1.2)
where Ω is a bounded domain in RN with C ∞ -boundary, gi ∈ C([0, ∞) × [0, ∞)),
gi (0, 0) < 0 and α, αi ∈ (0, 1), for i = 1, 2, 3.
Here, if α = αi = 0, for i = 1, 2, 3, the reaction terms are negative but finite.
Such problems are referred to as semipositone problems. (see [1, 3, 4, 6, 7, 8, 10,
2000 Mathematics Subject Classification. 35J25, 35J55.
Key words and phrases. Infinite semipositone; elliptic systems; combined non-linear effect.
c
2016
Texas State University.
Published March 21, 2016.
189
190
J. YE, J. ALI
EJDE-2016/CONF/23
11]). It is well documented in the literature that the study of positive solutions
to such semipositone problems are mathematically very challenging. Since the
test functions for positive subsolutions must come from positive functions ψ such
that −∆ψ < 0 near ∂Ω while −∆ψ > 0 in a large part of the interior of Ω (see
[5, 14]). In this paper, we study the more challenging semipositone problem where
the nonlinearities approach −∞ at the origin. Here we not only need to produce
subsolutions such that ψ > 0 in Ω, ψ = 0 on ∂Ω but also they must satisfy
limx→∂Ω (−∆ψ) = −∞. We refer to such problems as infinite semipositone systems.
We will seek positive solutions in [C 1 (Ω) ∩ C(Ω̄)]3 .
To state our results precisely we introduce the following hypotheses:
(H1) There exist σ > 0 and A > 0 such that α − α < σ < α and gi (s, t) > Asσ
for s 1, t 1, for i = 1, 2, 3 where α = max{α1 , α2 , α3 } and α =
min{α1 , α2 , α3 }.
(H2)
g1 (s, M g3 (s, s))
= 0, ∀M > 0.
lim
s→∞
s1+α1
(H3)
g2 (M g3 (s, s), s)
= 0, ∀M > 0.
lim
s→∞
s1+α2
(H4) There exist σ > 0 and A > 0 such that 0 < σ < α and gi (s, t) > Asσ for
s 1, t 1, for i = 1, 2, 3.
(H5)
g̃1 (s, M g̃3 (s, s))
lim
= 0, ∀M > 0.
s→∞
s
(H6)
g̃2 (M g̃3 (s, s), s)
lim
= 0, ∀M > 0,
s→∞
s
where g̃i (s, t) = gi (s, t)/sα .
We establish the following results.
Theorem 1.1. Assume (H1)–(H3) hold and gi (s, t) is nondecreasing in both variables for i = 1, 2, 3. Then system (1.1) has a positive solution for λ 1.
Theorem 1.2. Assume (H4)–(H6) hold and gi (s, t)/sα is nondecreasing in both
variables for i = 1, 2, 3. Then system (1.2) has a positive solution for λ 1.
We use the method of sub-super solutions to establish our results. Consider the
system
−∆u1 = λh1 (u1 , u2 , u3 ) in Ω
−∆u2 = λh2 (u1 , u2 , u3 )
in Ω
−∆u3 = λh3 (u1 , u2 , u3 )
in Ω
u1 = u2 = u3 = 0
on ∂Ω.
We define (ψ1 , ψ2 , ψ3 ) to be a subsolution of (1.3) if ψi ∈ C 1 (Ω) ∩ C(Ω̄) and
−∆ψi ≤ λhi (ψ1 , ψ2 , ψ3 )
ψi > 0
ψi = 0
in Ω
on ∂Ω,
in Ω
(1.3)
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POSITIVE SOLUTIONS
191
for i = 1, 2, 3, and (Z1 , Z2 , Z3 ) to be a supersolution of (1.3) if Zi ∈ C 1 (Ω) ∩ C(Ω̄)
and
−∆Zi ≥ λhi (Z1 , Z2 , Z3 )
Zi > 0
Zi = 0
in Ω
in Ω
on ∂Ω,
for i = 1, 2, 3. For systems (1.1) and (1.2), if there exist subsolutions (ψ1 , ψ2 , ψ3 )
and supersolutions (Z1 , Z2 , Z3 ) such that (ψ1 , ψ2 , ψ3 ) ≤ (Z1 , Z2 , Z3 ) on Ω̄, then
these systems have at least one solution (u1 , u2 , u3 ) ∈ [C 1 (Ω) ∩ C(Ω̄)]3 satisfying
(ψ1 , ψ2 , ψ3 ) ≤ (u1 , u2 , u3 ) ≤ (Z1 , Z2 , Z3 ) on Ω̄. This follows by the natural extension of the result in [9] for scalar equations to systems (1.1) and (1.2) under the
assumptions that gi (s, t)’s are nondecreasing and gis(s,t)
’s are nondecreasing in both
α
variables, respectively.
In [13], the authors study such singular systems in the case n = 2. (See also [15]
for a study in the case n = 1.) Here we extend this study to 3 × 3 bi-variate systems
(1.1) and (1.2). The main difference in these new systems is that our nonlinearities
depend on two variables instead of one variable, and this is more challenging in
constructing both sub and super solutions. We will prove Theorem 1.1 in Section 2
and Theorem 1.2 in Section 3. In Section 4, we will consider the natural extension
of our results to p-Laplacian systems.
2. Proof of main results
Theorem 1.1. Let φ > 0 such that kφk∞ = 1 be the eigenfunction corresponding
to the first eigenvalue of the operator −∆ with Dirichlet boundary condition, i.e.
φ satisfies
−∆φ = λ1 φ,
φ = 0,
For γ ∈
1
1
1+α , 1+(α−σ)
, let ψi = λγ φ
in Ω
on ∂Ω.
2
1+αi
. Then
i 2 −2α
1 − αi φ 1+αi λ1 φ2 −
|∇φ|2 .
1 + αi
1 + αi
Let δ > 0, m > 0 and µ > 0 be such that
1 − αi |∇φ|2 − λ1 φ2 ≥ m, in Ωδ , for i = 1, 2, 3,
1 + αi
−∆ψi = λγ
and φ ≥ µ > 0 in Ω \ Ωδ , where Ωδ = {x ∈ Ω | d(x, ∂Ω) ≤ δ}. This is possible since
|∇φ| =
6 0 on ∂Ω. Hence even though gi (0, 0) < 0, for λ 1, in Ωδ ,
2 1 − αi gi (0, 0)
(λγ
λ 1 φ2 −
|∇φ|2 ≤ λ γ αi ,
1 + αi
1 + αi
(λ )
since 1 − γ − αi γ < 0. Therefore,
− ∆ψi ≤ λ
gi (0, 0)
(λγ φ
2
1+αi
≤λ
)αi
gi (ψi+1 , ψi+2 )
ψiαi
in Ωδ
for λ 1.
Next, in Ω \ Ωδ , since φ ≥ µ > 0, from (H1), we know that for λ 1,
2
2
2
gi (λγ φ 1+αi+1 , λγ φ 1+αi+2 ) ≥ A(λγ φ 1+αi+1 )σ .
(2.1)
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Also, since 0 < µ ≤ φ < 1 and 1 + (σ − αi )γ − γ > 0, for λ 1,
2
2 A(λγ φ 1+αi+1 )σ
λ
λ1 φ2 ≤ λ
.
1 + αi
λγαi
γ
Then in Ω \ Ωδ , for λ 1,
−∆ψi ≤ λγ
−2αi
2 +2
λ1 φ 1+αi
1 + αi
2
≤λ
2
gi (λγ φ 1+αi+1 , λγ φ 1+αi+2 )
2
(λγ φ 1+αi )αi
gi (ψi+1 , ψi+2 )
.
=λ
ψiαi
(2.2)
Combining (2.1) and (2.2), we see that for λ 1,
−∆ψi ≤ λ
gi (ψi+1 , ψi+2 )
ψiαi
in Ω.
Thus (ψ1 , ψ2 , ψ3 ) is a positive subsolution of (1.1).
Now, we construct a supersolution (Z1 , Z2 , Z3 ) ≥ (ψ1 , ψ2 , ψ3 ). From [12], we
know that wi ∈ C 1 (Ω) ∩ C(Ω) exists such that
1
−∆wi = αi , in Ω,
wi
wi = 0, on ∂Ω,
and satisfying wi ≥ εe for some ε > 0. Here e is a positive solution of −∆e = 1 in
∂e
Ω and e = 0 on ∂Ω which satisfies e ∈ C01 (Ω) and ∂ν
< 0 on ∂Ω, where ν is the
outward normal vector on ∂Ω. Let ω = max{kw1 k, kw2 k, kw3 k}, and
(Z1 , Z2 , Z3 ) = (m(λ)w1 , m(λ)w2 , g3 (m(λ)kw1 k, m(λ)kw2 k)w3 ).
Then, from (H2), we can choose m(λ) 1 such that
1
g1 (m(λ)w, g3 (m(λ)w, m(λ)w)w)
≤ .
(m(λ))1+α1
λ
Then
−∆Z1 =
m(λ)
g1 (m(λ)w, g3 (m(λ)w, m(λ)w)w)
≥λ
w1α1
(m(λ)w1 )α1
g1 (m(λ)w2 , g3 (m(λ)kw1 k, m(λ)kw2 k)w3 )
≥λ
(m(λ)w1 )α1
g1 (Z2 , Z3 )
=λ
.
Z1α1
From (H3), choose m(λ) 1 such that
g2 (g3 (m(λ)w, m(λ)w)w, m(λ)w)
1
≤ .
(m(λ))1+α2
λ
Then
−∆Z2 =
m(λ)
g2 (g3 (m(λ)w, m(λ)w)w, m(λ)w)
≥λ
w2α2
(m(λ)w2 )α2
g2 (g3 (m(λ)kw1 k, m(λ)kw2 k)w3 , m(λ)w1 )
≥λ
(m(λ)w2 )α2
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POSITIVE SOLUTIONS
=λ
193
g2 (Z3 , Z1 )
.
Z2α2
From (H1), choose m(λ) 1 such that
λ
< 1.
(g3 (m(λ)kw1 k, m(λ)kw2 k))α3
Then
g3 (m(λ)kw1 k, m(λ)kw2 k)
w3α3
g3 (m(λ)w1 , m(λ)w2 )
≥λ
(g3 (m(λ)kw1 k, m(λ)kw2 k))α3 w3α3
g3 (Z1 , Z2 )
=λ
.
Z3α3
−∆Z3 =
Thus (Z1 , Z2 , Z3 ) is a supersolution of (1.1). Further, m(λ) can be chosen large
enough so that (Z1 , Z2 , Z3 ) ≥ (ψ1 , ψ2 , ψ3 ) in Ω. Therefore, problem (1.1) has a
positive solution (u1 , u2 , u3 ) ∈ [(ψ1 , ψ2 , ψ3 ), (Z1 , Z2 , Z3 )].
2
1
1
, 1+(α−σ)
, and φ as before.
Proof of Theorem 1.2. Let ψ = λγ φ 1+α , γ ∈ 1+α
Then by arguments similar to that in the proof of Theorem 1.1, we can show
that (ψ, ψ, ψ) is a subsolution. Now, we construct a supersolution (Z1 , Z2 , Z3 ) ≥
(ψ, ψ, ψ). From (H5), (H6), we can choose m(λ) 1 such that
1
g̃1 (m(λ)kek, λg̃3 (m(λ)kek, m(λ)kek)kek)
≤ ,
m(λ)
λ
g̃2 (λg̃3 (m(λ)kek, m(λ)kek)kek, m(λ)kek)
1
≤ ,
m(λ)
λ
where e is as described before in the proof of Theorem 1.1. Let
(Z1 , Z2 , Z3 ) := (m(λ)e, m(λ)e, λg̃3 (m(λ)kek, m(λ)kek)e).
Then by (2.3)
−∆Z1 = m(λ) ≥ λg̃1 (m(λ)kek, λg̃3 (m(λ)kek, m(λ)kek)kek)
g̃1 (m(λ)e2 , λg̃3 (m(λ)kek, m(λ)kek)e)
(m(λ)e)α
g1 (Z2 , Z3 )
,
=λ
Z2α
≥λ
and by (3.2)
−∆Z2 = m(λ) ≥ λg̃2 (λg̃3 (m(λ)kek, m(λ)kek)e, m(λ)kek)
g2 (λg̃3 (m(λ)kek, m(λ)kek)e, m(λ)e)
(λg̃3 ((m(λ)kek, m(λ)kek)e)α
g2 (Z3 , Z1 )
=λ
,
Z3α
≥λ
and
−∆Z3 = λg̃3 (m(λ)kek, m(λ)kek)
≥ λg̃3 (m(λ)e, m(λ)e)
(2.3)
(2.4)
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g3 (m(λ)e, m(λ)e)
(m(λ)e)α
g3 (Z1 , Z2 )
.
=λ
Z1α
=λ
Thus (Z1 , Z2 , Z3 ) is a supersolution of (1.2). Further, m(λ) can be chosen large
enough so that (Z1 , Z2 , Z3 ) ≥ (ψ1 , ψ2 , ψ3 ) in Ω. Therefore, problem (1.2) has a
positive solution (u1 , u2 , u3 ) ∈ [(ψ1 , ψ2 , ψ3 ), (Z1 , Z2 , Z3 )].
3. p-Laplacian systems
In this section, we discuss the extensions of our main results to the following two
p-Laplacian systems:
g1 (u2 , u3 )
, in Ω,
−∆p u1 = λ
1
uα
1
g2 (u3 , u1 )
−∆p u2 = λ
, in Ω,
2
uα
(3.1)
2
g3 (u1 , u2 )
−∆p u3 = λ
, in Ω,
3
uα
3
u1 = u2 = u3 = 0, on ∂Ω,
and
g1 (u2 , u3 )
, in Ω,
uα
2
g2 (u3 , u1 )
−∆p u2 = λ
, in Ω,
uα
3
g3 (u1 , u2 )
, in Ω,
−∆p u3 = λ
uα
1
u1 = u2 = u3 = 0, on ∂Ω.
−∆p u1 = λ
(3.2)
Here ∆p u = div(|∇u|p−2 ∇u), Ω is a bounded domain in RN with C ∞ -boundary,
gi ∈ C([0, ∞) × [0, ∞)), gi (0, 0) < 0 and α, αi ∈ (0, 1), for i = 1, 2, 3.
To state our results for these p-Laplacian systems, we introduce the following
hypotheses:
(H7)
1
g1 (s, M (g3 (s, s)) p−1 )
lim
= 0,
s→∞
sp−1+α1
∀M > 0.
(H8)
1
g2 (M (g3 (s, s)) p−1 , s)
= 0,
s→∞
sp−1+α2
lim
∀M > 0.
(H9)
1
g̃1 (s, M (g̃3 (s, s)) p−1 )
= 0,
s→∞
sp−1
lim
∀M > 0.
(H10)
1
g̃2 (M (g̃3 (s, s)) p−1 , s)
lim
= 0,
s→∞
sp−1
α
where g̃i (s, t) = gi (s, t)/s .
∀M > 0,
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POSITIVE SOLUTIONS
195
Theorem 3.1. Assume (A) p ≥ 3 or (B) p < 3 and αi < p3 . Let (H1), (H7), (H8)
hold and gi (s, t) be nondecreasing in both variables for i = 1, 2, 3. Then system
(3.1) has a positive solution for λ 1.
Theorem 3.2. Assume (H4), (H9), (H10) hold and gi (s, t)/sα is nondecreasing in
both variables for i = 1, 2, 3. Then system (3.2) has a positive solution for λ 1.
Here we prove these results again by the method of sub-super solutions. As
described in [13], the method of sub-super solutions holds for systems (3.1) and (3.2)
with the assumptions that gi (s, t)’s are nondecreasing and the functions gi (s, t)/sα
are nondecreasing in both variables. First, by an argument similar to the proof of
p
p−1+α
i
, for
Theorem 1.1, we can show that if ψi := λγ φp
1
1
,
γ∈(
),
p − 1 + α p − 1 + (α − σ)
then (ψ1 , ψ2 , ψ3 ) is subsolution of (3.1) for λ 1. Here φp > 0 such that kφp k∞ = 1
is the eigenfunction corresponding to the first eigenvalue of the operator −∆p with
Dirichlet boundary condition, i.e. φp satisfies:
−∆p φp = λ1 φpp−1 ,
φp = 0,
in Ω
on ∂Ω.
Also, by [2], for (A) p ≥ n, or (B) p < n and αi <
p
n,
the problem
1
, in Ω
wiαi
wi = 0, on ∂Ω,
−∆p wi =
has a solution wi ∈ C 1 (Ω) × C(Ω) such that wi ≥ εep , where −∆p ep = 1 in Ω,
ep = 0 on ∂Ω. Let (Z1 , Z2 , Z3 ) := (m(λ)w1 , m(λ)w2 , g3 (m(λ)kw1 k, m(λ)kw2 k)w3 ).
Then for m(λ) 1, (Z1 , Z2 , Z3 ) is a supersolution of (3.1) and (Z1 , Z2 , Z3 ) ≥
(ψ1 , ψ2 , ψ3 ), by an argument similar to that in the proof of Theorem 1.1. Hence
Theorem 3.1 holds.
p
1
1
Next, to establish theorem 3.2, let ψ := λγ φpp−1+α , for γ ∈ ( p−1+α
, p−1+(α−σ)
),
and
1
(Z1 , Z2 , Z3 ) := (m(λ)ep , m(λ)ep , λ p−1 g̃3 (m(λ)kep k, m(λ)kep k)ep ).
Then by an argument similar to that in the proof of Theorem 1.2, (ψ, ψ, ψ) is a
subsolution of (3.2) for λ 1 and for m(λ) 1, (Z1 , Z2 , Z3 ) is a supersolution of
(3.2) with (Z1 , Z2 , Z3 ) ≥ (ψ, ψ, ψ). Hence Theorem 3.2 holds.
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Jinglong Ye
Department of Mathematics, Computer and Information Sciences, Mississippi Valley
State University, Itta Bena, MS 38941, USA
E-mail address: jinglong.ye@mvsu.edu
Jaffar Ali
Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL 33965,
USA
E-mail address: jahameed@fgcu.edu