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Synthesis, Characterisation, Interaction with DNA, Cytotoxicity, and Apoptotic Studies of Ruthenium(ii) Polypyridyl Complexes
Xie and Ma Advances in Difference Equations (2017) 2017:10
DOI 10.1186/s13662-016-1071-4
RESEARCH
Open Access
The dynamics of a diffusive logistic model
with nonlocal terms
Xianhua Xie* and Li Ma
*
Correspondence:
xxianhua@sina.com.cn
Key Laboratory of Jiangxi Province
for Numerical Simulation and
Emulation Techniques, College of
Mathematics and Computers,
Gannan Normal University,
Ganzhou, Jiangxi 341000, People’s
Republic of China
Abstract
It is well known that the set of positive solutions may contain crucial clues for the
stationary patterns. In this paper, we consider a class of diffusive logistic equations
with nonlocal terms subject to the Dirichlet boundary condition in a bounded
domain. We study the existence of positive solutions under certain conditions on the
parameters by using bifurcation theory. Finally, we illustrate the general results by
applications to models with one-dimensional spatial domain.
MSC: Primary 35J15; secondary 92B20
Keywords: logistic model; positive steady-state solutions; a priori bounds;
bifurcation theory
1 Introduction
Recently, many researchers pay more attention on the studies of reaction-diffusion equations; we refer to, for example, [–]. Ecologically, positive solutions correspond to the
existence of steady states of species. It is well known that the set of positive solutions may
contain crucial clues for the stationary patterns. From the mathematical viewpoint, it is
important to derive some information about the set of positive solutions by means of the
coefficients such as the growth rate of the species. Especially, most of the references concentrated on diffusive models with a single population (see, e.g., [, , ]). One of the most
classical diffusive logistic equations is
ut – u = λu( – K(x)up ) in ,
u=
on ∂,
(.)
which was regarded as a logistic system of individual species in the ecological studies.
Here, u(x) is the population density at location x ∈ , λ ∈ R+ is the growth rate of the
species and is usually deemed to be a variable, K is a positive function denoting the carrying capacity, and p > . In (.), we assume that is surrounded by inhospitable areas,
subjected to the homogeneous Dirichlet boundary conditions.
Later, many scientists found that the movement of an individual species is sometimes determined by surrounding conditions around the point where the species stays. For example, we consider movements of animals, where each individual species mutually interacts
by seeing, hearing, and smelling around themselves. That is why interaction by chemical
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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indicate if changes were made.
Xie and Ma Advances in Difference Equations (2017) 2017:10
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means may take place under certain circumstances. Hence, it seems more realistic to take
account of nonlocal effects in the study of species dynamics; see [, , , , , ]. Usually,
this nonlocal effect depends on the value of the population around x, that is, the crowding effect depends on the series of values of u. In some special cases, this nonlocal effect
also depends on the value in a neighborhood Br (x) of x, where Br (x) represents the ball
centered at x of radius r > . Along these reasons, system (.) is replaced by the following
more general diffusive logistic population models with nonlocal effect:
ut – u = λu( – K(x, y)up (y) dy)
u=
in ,
on ∂,
(.)
and
ut – u = u(λ – ∩Br (x) K(y)up (y) dy)
u=
in ,
on ∂,
(.)
where p > , and K : × → R and K : → R are nonnegative and nontrivial continuous functions. Chen and Shi [] considered the dynamical behavior of system (.) when
p = and the kernel function K(x, y) is a continuous and nonnegative function on ×
satisfying K(x, y)u(y) dy > for all positive continuous functions u on . Applying the
implicit function theorem, Chen and Shi [] obtained the existence and uniqueness of a
positive steady-state solution of system (.) when < λ – λ , where λ denotes the
first eigenvalue of the minus Laplacian operator under homogeneous Dirichlet boundary
conditions. Some researchers [, , , ] also realized that the kernel function K(x, y) may
have no direct connection with the growth rate λ of the species. For example, Allegretto
and Nistri [] studied the following model:
–u = u(λ – K(x, y)up (y) dy)
u=
in ,
on ,
(.)
where K(x, y) vanishes away from the diagonal domain of RN × RN . Allegretto and Nistri
[] found that (.) possesses a unique positive solution when λ > λ if K(x, y) = Kδ (|x – y|)
is a mollifier in RN , that is, Kδ (|x – y|) ∈ C∞ , RN Kδ (|x – y|) dy = for any x with
Kδ |x – y| = when |x – y| ≥ δ,
and Kδ (|x – y|) is bounded away from zero when |x – y| < μ < δ. Later, Corrêa et al. []
proved that (.) possesses a unique positive solution if K(x, y) is a separable variable, that
is, K(x, y) = g(x)h(y), where h ≥ , h = , and g(x) > in . Sun et al. [] investigated
the existence of positive solutions of system (.) with K(x, y) = K (|x – y|) and = (–, ),
where K : [, ] → (, ∞) is a nondecreasing and piecewise continuous function satisfying
K (y) dy > . Besides, Alves et al. [] also studied the existence of a positive solution of
system (.).
In the aforementioned literature, the authors only concentrated on the single species.
For the model with two populations, in particular, for those diffusive Lotka-Volterra systems without nonlocal terms, the questions posed have been extensively studied in [, ,
Xie and Ma Advances in Difference Equations (2017) 2017:10
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] and references therein. However, the discussion of the dynamical behavior of two interacting species in the presence of nonlocal term effects is more difficult than those models
without nonlocal term effects. Lately, Guo and Yan [] employ Lyapunov-Schmidt reduction to investigate the existence of the positive solution of the following model:
⎧
⎪
⎨ –u = λu( – A (x, y)u(y) dy – A (x, y)v(y) dy)
–v = λv( – A (x, y)u(y) dy – A (x, y)v(y) dy)
⎪
⎩
u=v=
in ,
in ,
on ,
(.)
where u(x) and v(x) are the population densities at location x, λ > is a scaling constant,
and is a connected bounded open domain in RN (N ≥ ) with a smooth boundary ∂.
The kernel functions Aij (i, j = , ) describe the dispersal behaviors of the populations.
A natural problem is whether (.) has a positive solution for λ not only near to but also
far away from λ . Moreover, it is very interesting to investigate the following more general
population model with nonlocal delay effect:
⎧
p
q
⎪
⎨ –u = u(λ – A (x, y)u (y) dy – A (x, y)v (y) dy)
–v = v(λ – A (x, y)up (y) dy – A (x, y)vq (y) dy)
⎪
⎩
u=v=
in ,
in ,
on ,
(.)
where ⊂ RN (N ≥ ) is a bounded domain with a smooth boundary ∂, p, q are positive
constants, and Aij ∈ L∞ ( × , R), i, j = , . Here u(x) and v(x) can be interpreted as the
densities of prey and predator populations at a spatial position x ∈ , and the parameter
λ is a positive real number representing the growth rate of the prey and predator.
The purpose of this paper is to find sufficient conditions ensuring the existence of a
positive solution for all λ > λ . Our main approach is global bifurcation theory, which is
different from the method adopted in []. Moreover, we also obtain the stability of the
positive solution by analyzing the distribution of the eigenvalues, which was not considered by Alves et al. []. Throughout this paper, we impose the following assumptions on
the dispersal kernel functions Aij (x, y), i, j = , .
(C) T and T are positive on the space C + () × C + () in the sense that
Ti (C + () × C + ()) ⊂ C + () × C + () \ {(, )}, i = , , where C + () represents
the space of positive continuous functions, and
Aj (·, y)up (y) + Aj (·, y)vq (y) dy,
Tj (u, v) =
j = , .
(C) If u, v are measurable and satisfy
u(y) p
|v(y)|q u(x)
× [A (x, y)| u | + A (x, y) u p ]( u ) dy dx = ,
|u(y)|p
v(y) q v(x)
× [A (x, y) v q + A (x, y)| v | ]( v ) dy dx = ,
then u = v = a.e. in . Here we set
norm in H (), that is,
u = u H () =
|∇u| dx,
u
u
= for u = and denote by · the usual
Xie and Ma Advances in Difference Equations (2017) 2017:10
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where the space H () = {U ∈ H () | U(x) = , ∀x ∈ ∂} and H k () (k ≥ ) is
dn f
the Sobolev space of L -functions f on with derivatives dx
n (n = , , . . . , k)
belonging to L ().
Our main results are stated as follows.
Theorem . Suppose that Aij (i, j = , ) satisfy (C) and (C). Then problem (.) has a
positive solution if and only if λ > λ .
In view of Theorem ., we see that system (.) with Aij (x, y) = Aij (x, y)χBr (x) (i, j = , )
has a positive solution if and only if λ > λ , where Aij (x, y) (i, j = , ) are positive functions
on × . If Aij (i, j = , ) do not satisfy assumption (C), then Aij (i, j = , ) may vanish
in some neighborhood of the diagonal of × (see Section ). In this case, Theorem .
is inapplicable. However, we are also able to investigate the existence and nonexistence of
positive solutions for some value of λ under the following assumptions on the dispersal
kernel functions Aij (x, y), i, j = , .
(C) There are r > and m connected open sets , , . . . , m ⊂ such that
i ∩ j = ∅, i = j, and Aij (x, y) > for all (x, y) ∈ × satisfying x ∈/ m
j= j and
|x – y| < r, i, j = , .
In view of [], we know that if ⊂ , then λ ( ) ≥ λ ( ). Moreover, the inequality
is strict as soon as \ contains a set of positive capacity (since the first eigenfunction
cannot vanish on such a set). Hence, we have the following result.
Theorem . Suppose that Aij (i, j = , ) satisfy (C) and (C). Then, problem (.) has a
positive solution when λ < λ < min{λ ( ), . . . , λ (m )}, where λ (i ) denotes the principal
eigenvalue of the minus Laplacian operator in i under homogeneous Dirichlet boundary
conditions, i = , , . . . , m. Moreover, the solution is stable.
The remaining parts of the paper are structured in the following way. In Section , we
employ the global bifurcation theory to obtain the existence and stability of positive solutions of (.) under conditions (C) and (C). Section is devoted to the case where Aij
(i, j = , ) satisfy condition (C). Section is devoted to the application of our theoretical
results to some one-dimensional models.
2 Proof of Theorem 1.1
ij
In this section, we introduce some basic results. First, consider the functions φp,ω : → R
(i, j = , ) given by
ij
φp,ω
(x) =
p
Aij (x, y)ω(y) dy,
x ∈ .
ij
If Aij (i, j = , ) and ω are bounded, then φp,ω (i, j = , ) are well defined. Moreover, we
have the following observations:
ij
φ ≤ Aij ∞ || ω p for all ω ∈ L∞ (),
p,ω ∞
∞
ij
φ – φ ij ≤ Aij ∞ |||ω|p – |ν|p
for all ω, ν ∈ L∞ (),
p,ω
p,ν ∞
∞
(.)
(.)
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and
φpij : L∞ () → L∞ (),
ij
φpij (u) = φp,u
(i, j = , ) are uniformly continuous in L∞ ().
(.)
Using these notations, it is easy to observe that (u, v) is a positive solution of (.) if and
only if (u, v) is a positive solution of
⎧
⎪
⎨ –u = u(λ – φp,u – φq,v ) in ,
(.)
–v = v(λ – φp,u – φq,v ) in ,
⎪
⎩
u=v=
on ∂.
First, we show the nonexistence of a positive solution of (.) for small λ.
Lemma . Suppose that Aij (i, j = , ) satisfy (C) and (C). Then system (.) with λ < λ
has no positive solutions.
Proof We prove this lemma by contradiction. Assume that (.) with λ ≤ λ has a positive
solution (u∗ , v∗ ). Then we have
⎧
p
q
⎪
⎨ –u∗ = u∗ (λ – A (x, y)u∗ (y) dy – A (x, y)v∗ (y) dy) in ,
p
q
(.)
–v∗ = v∗ (λ – A (x, y)u∗ (y) dy – A (x, y)v∗ (y) dy) in ,
⎪
⎩
on ∂.
u∗ = v∗ =
Let (λ , ψ ) with ψ > be the principle eigenpair of the eigenvalue problem
–ψ = μψ in ,
ψ =
on ∂.
(.)
Multiplying (.) by ψ and then integrating it on , we have
⎧
⎪
(λ – λ )ψ (x)u∗ (x) dx
⎪
⎪
p
q
⎨ =
× [A (x, y)u∗ (y) + A (x, y)v∗ (y)]u∗ (x)ψ (x) dy dx,
⎪
(λ – λ )ψ (x)v∗ (x) dx
⎪
⎪
⎩
p
q
= × [A (x, y)u∗ (y) + A (x, y)v∗ (y)]v∗ (x)ψ (x) dy dx.
(.)
Since ψ > , u∗ > , v∗ > , and λ ≤ λ , we find that each of the left-hand sides of the two
equations of (.) is less than and that each of the right-hand sides of the two equations
is greater than , which is a contradiction. So system (.) has no positive solution for
λ ≤ λ .
Proposition ([]) Assume that there exists a pair of positive functions u, v ∈ C () ∩
C,δ (), δ ∈ (, ), such that
⎧
p
q
–u – u[λ – A (x, y)u (y) dy – A (x, y)v (y) dy]
⎪
⎪
⎪
p–
q–
⎨
+ u [pA (x, y)u (y)u(y) + qA (x, y)v (y)v(y)] dy > ,
p
q
⎪
– v[λ – A (x, y)u (y) dy – A (x, y)v (y) dy]
–v
⎪
⎪
⎩
p–
q–
+ v [pA (x, y)u (y)u(y) + qA (x, y)v (y)v(y)] dy > .
Then the principle eigenvalue of the eigenvalue problem (.) is positive.
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In the following section, we intend to prove the existence of a positive solution for (.)
by using the classical bifurcation result of Rabinowitz []. To this end, we recall that there
exists c∞ = c∞ () > such that, for each f ∈ L∞ (), there exists a unique ω ∈ C () satisfying
–ω = f (x), x ∈ ,
ω = ,
x ∈ ∂,
(.)
and ω C () ≤ c∞ f ∞ . Thus, the solution operator S : C () → C () can be given by
SU = ω
⇔
–ω = U, x ∈ ,
x ∈ ∂.
ω = ,
Obviously, S is well defined, linear, and satisfies
SU C () ≤ c∞ U C () ,
∀U ∈ C ().
Moreover, by the Schauder imbedding theorem, S : C () → C () is a compact operator.
In view of the spectrum of S, it is easy to see that
σ (S) = λ–
j | λj is an eigenvalue of the minus Laplacian operator .
On the other hand, define the nonlinear operator F : C () → C () as
FV = ω
⇔
–ω +
ω = ,
V V = ,
x ∈ ,
x ∈ ∂,
where V = (u, v)T ∈ C () and
φp,u
+ φq,v
.
V =
φp,u
+ φq,v
Obviously, F is continuous and satisfies
FV C () ≤ c∞
V ∞
V C () ,
∀V ∈ C (),
Using again the Schauder imbedding theorem, we see that F : C () → C () is compact.
Furthermore, note that
FV C () ≤ FV C () .
Then we have
FV
V
C ()
C ()
≤
FV C ()
V C ()
≤ c∞
V ∞,
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from which it follows that
lim
V →
FV
V C ()
= ,
i.e.,
FV = o V C () .
(.)
Obviously, V = (u, v)T solves (.) if and only if
V = G(λ, V ) λSV + FV .
In view of [], considering E = C (), we have the following result.
Theorem . Let E be a Banach space. Suppose that F satisfies (.), S is a compact linear
operator, and λ– ∈ σ (S) with odd algebraic multiplicity. Let
= (λ, V ) ∈ R × E : V = λSV + FV , V = ,
and let C be a closed connected component of that contains (λ, ). Then either
() C is unbounded in R × E, or else
() there exists λ̃ = λ such that (λ̃, ) ∈ C and λ̃– ∈ σ (S).
Remark . Because λ is the principle eigenvalue of the eigenvalue problem (.) with
the associated eigenfunction ψ > on and its multiplicity is simple, by the global bifurcation theorem there exists a closed connected component C containing (λ , ) and
satisfying () or () for solutions to (.).
In order to prove the existence of positive solutions of (.) with λ > λ , it follows from
Lemma . and Theorem . that it only suffices to prove that conclusion () of Theorem . holds and that V is bounded when λ > λ .
Lemma . There exists ε > such that if (λ, V ) = (λ, u, v) ∈ C with λ – λ < ε and
V C () < ε where u = and v = , then u and v have definite signals, that is,
(i) u(x) > and v(x) > for all x ∈ , or
(ii) u(x) > and v(x) < for all x ∈ , or
(iii) u(x) < and v(x) > for all x ∈ , or
(iv) u(x) < and v(x) < for all x ∈ .
Proof Take Vn = (un , vn ) ∈ C () and λn → λ as n → ∞ such that Vn C (×) → as
n → ∞ and
Vn = G(λn , Vn ).
Let wn = un un
C ()
and wn = vn vn
C ()
. Then we have
⎧
⎪
⎨ –wn + φp,un wn + φq,vn wn = λn wn
–wn + φp,un wn + φq,vn wn = λn wn
⎪
⎩
wn = wn =
in ,
in ,
on ∂.
(.)
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It follows from (.) that φp,u
, φq,v
, φp,u
, and φq,v
are bounded if both
n ∞
n ∞
n ∞
n ∞
un and vn are bounded in C (). Thus, it is easy to see that
w ≤ c∞ λn + φ + φ w ,
n C ()
p,un ∞
q,vn ∞
n C ()
w ≤ c∞ λn + φ + φ w ,
n C ()
p,un ∞
q,vn ∞
n C ()
∀n ∈ N,
∀n ∈ N.
Note that
w – w ≤ c∞ λn w – λm w + φ + φ w – w
n
m C ()
n
m C ()
p,un ∞
q,vn ∞
n
m C ()
, ∀n ∈ N,
+ φp,u
– φp,u
+ φq,v
– φq,v
n
m ∞
n
m ∞
w – w ≤ c∞ λn w – λm w + φ + φ w – w
n
m C ()
n
m C ()
p,un ∞
q,vn ∞
n
m C ()
, ∀n ∈ N.
+ φp,u
– φp,u
+ φq,v
– φq,v
n
m ∞
n
m ∞
Then, using the Arzelà-Ascoli theorem, we see that, for each fixed i ∈ {, }, win converge
to some wi ∈ C () uniformly in , and hence there exists a convergent subsequence.
By the definition of win , wi C () = implies wi = , i = , . Multiplying (.) by v and
integrating on , we have
∇wn ∇v dx +
∇wn ∇v dx +
φp,un + φq,v
wn v dx = λn
n
φp,un + φq,v
wn v dx = λn
n
wn v dx,
wn v dx.
w , φq,v
w , φp,u
w , φq,v
w → as n → ∞ in C (). Then
In view of (.), we have φp,u
n n
n n
n n
n n
⎧
⎪
⎨ –w = λ w
–w = λ w
⎪
⎩
w = w =
in ,
in ,
on ∂.
Since w w = , it follows from the spectral and limit theory that, for each fixed i ∈ {, },
wi (x) > or wi (x) <
for all x ∈ . In what follows, we only consider the case where w (x) > and w (x) > for
all x ∈ because the other three cases can be discussed analogously. Note that w and w
are the C ()-limits of wn and wn , respectively. Then, on , win > and win > for n large
enough. Thus, the signs of un and vn are the same as those of wn and wn for n large enough.
This completes the proof.
It is easy to check that if (λ, u, v) ∈ , then the pairs (λ, –u, v), (λ, u, –v), and (λ, –u, –v)
are also in . In what follows, we decompose C into C = C +,+ ∪ C +,– ∪ C –,+ ∪ C –,– , where
C +,+ = (λ, u, v) ∈ C : u(x) ≥ , v(x) ≥ , ∀x ∈ ;
C +,– = (λ, u, v) ∈ C : u(x) ≥ , v(x) ≤ , ∀x ∈ ;
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C –,+ = (λ, u, v) ∈ C : u(x) ≤ , v(x) ≥ , ∀x ∈ ;
C –,– = (λ, u, v) ∈ C : u(x) ≤ , v(x) ≤ , ∀x ∈ .
The following lemma tells the fact that system (.) satisfies Theorem .().
Lemma . Each of C +,+ , C –,+ , C –,+ , and C –,– is unbounded.
Proof It is easy to see that if one of C +,+ , C –,+ , C –,+ , and C –,– is unbounded then the others
are also unbounded. Therefore, it suffices to show that C +,+ is unbounded. Suppose that
C +,+ is bounded. Then C is also bounded. In view of the global bifurcation theorem (Theorem .), C satisfies conclusion () of Theorem ., that is, C contains (λ̃, , ), where λ̃ = λ
and λ̃– ∈ σ (S).
+,+
such that un vn = and (un , vn ) = G(λn , un , vn ) and such
We take {(λn , un , vn )}∞
n= ⊆ C
that λn → λ̃, un C () → , and un C () → as n → ∞. Letting
wn =
un
un C ()
wn =
,
vn
vn C ()
and using similar arguments as in the proof of Lemma ., we get that (wn , wn ) converges
to (w , w ) in C () × C () as n → ∞, which is a nonzero solution pair of the eigenvalue
problem
⎧
⎪
⎨ –w = λ̃w
–w = λ̃w
⎪
⎩
w = w =
in ,
in ,
on ∂,
which implies that both w and w are eigenfunctions associated with λ̃. Since λ̃ = λ , both
w and w change signs in . Thus, for n large enough, wn and wn change signs, and hence
the same results hold for un = un C () wn and vn = vn C () wn . However, this contradicts
the assumption that (λn , un , vn ) ∈ C +,+ . This completes the proof.
Now, we shall prove that system (.) satisfies Theorem .(). It suffices to show that the
connected component C +,+ intersects any set of the form {λ} × H () × H () for λ > λ .
Lemma . Suppose that Aij (i, j = , ) satisfy conditions (C) and (C). For any > ,
there exists a constant r > such that u C () ≤ r and v C () ≤ r whenever (λ, u, v) ∈ C +,+
and λ ≤ .
Proof First, we denote by · the usual norm in H (), that is,
u = u H () =
|∇u| dx.
Indeed, if this were not true, there would exist {(λn , un , vn )}∞
n= ⊂ [, ] × H () × H ()
such that
Case . un → ∞, λn → λ ∈ as n → ∞, vn ≤ r, and (un , vn ) = G(λn , un , vn );
Case . vn → ∞, λn → λ ∈ as n → ∞, un ≤ r, and (un , vn ) = G(λn , un , vn );
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Case . un → ∞, vn → ∞ and λn → λ ∈ as n → ∞, and (un , vn ) = G(λn , un , vn ).
We only discuss Case because the other cases can be dealt with analogously. Let
wn =
un
,
un C ()
wn =
vn
.
vn C ()
Then we have
∇wn ∇θ dx + (φp,un + φq,vn )wn θ dx = λn wn θ dx,
∇wn ∇θ dx + (φp,un + φq,vn )wn θ dx = λn wn θ dx.
(.)
Note that {(wn , wn )}∞
n= are bounded in H () × H (). Then, without loss of generality,
we suppose that there is w = (w , w ) ∈ H () × H () such that
win → wi
as n → ∞ in H (), i = , ,
win → wi
as n → ∞ in L (), i = , ,
and win (x) → wi (x) as n → ∞ a.e. in , i = , . Taking θ = u unp+ as a test function in the
n
first equation of (.) and θ = v vnq+ as a test function in the second equation of (.),
n
respectively, equations (.) reduce to
⎧
⎪
)(wn ) dx = uλnn p (wn ) dx,
vn
⎨ un p + (φp,w
+ φq,
n
un p/q
λn
⎪
⎩ vn q + (φp, un + φq,w )(wn ) dx = vn q (wn ) dx.
n
vn q/p
Passing to the limit in these equalities and using the Fatou lemma, we have
⎧
⎪
)(w ) dx
≤ (φp,w
v
+ φq,
⎪
⎪
u p/q
⎪
⎪
⎪ ≤ lim
⎪
⎨
n→∞ (φp,w + φq, vn )(wn ) dx = ,
n
un p/q
⎪
≤ (φp, u + φq,w
⎪
)(w ) dx
⎪
q/p
⎪
v
⎪
⎪
⎪
⎩ ≤ limn→∞ (φp, un + φq,w
)(wn ) dx = .
vn q/p
n
Take into account condition (C), we have w = w = . Namely, wn and wn converge to
in L (). On the other hand, taking θ = wn and θ = wn as test functions in (.), we see
that
|∇wn | dx + (φp,un + φq,vn )(wn ) dx = λn (wn ) dx,
|∇wn | dx + (φp,un + φq,vn )(wn ) dx = λn (wn ) dx.
Note that {λn }∞
n= is bounded from above by and
φp,un + φq,v
wn dx ≥ ,
n
φp,un + φq,v
wn dx ≥ .
n
Xie and Ma Advances in Difference Equations (2017) 2017:10
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Then we have
|∇wn | dx ≤ λn (wn ) dx,
|∇wn | dx ≤ λn (wn ) dx.
Taking the limit, we conclude that wn → and wn → as n → ∞, which is absurd
because wn = wn = for all n. This completes the proof.
3 Proof of Theorem 2.1
We observe that if Aij (i, j = , ) do not satisfy condition (C), that is:
(C) There exists a measurable function ϕ = (ϕ , ϕ ) : → R \ {(, )} such that
ϕ
φp, ϕ + φq, ϕ
dx = or
ϕ
ϕ
ϕ p/q
ϕ
φp, ϕ + φq, ϕ
dx = .
ϕ
ϕ
ϕ q/p
(.)
Lemma . If Aij (i, j = , ) satisfy condition (C), then (.) implies that ϕ = ϕ = a.e.
in \ U.
Proof In view of equation (.), we have
φp, ϕ + φq, ϕ
ϕ p/q
ϕ
ϕ
ϕ
= a.e. in ,
or
φp,
ϕ
ϕ q/p
+ φq, ϕ
ϕ
ϕ
ϕ
=
a.e. in .
Fixing ε > and Dε = {x ∈ \ U : |ϕ | ≥ ε or |ϕ | ≥ ε}, it follows that
ϕ
= φp, ϕ + φq, ϕ
ϕ
ϕ
ϕ p/q
p
q
ϕ (y)
+ A (x, y) |ϕ (y)| dy
A (x, y)
=
ϕ
ϕ p
p
εq
ε
≥ min
A (x, y)
dy,
A (x, y) dy ,
p
ϕ
Dε ∩Br
Dε ∩Br ϕ
or
= φp,
=
ϕ
ϕ q/p
+ φq, ϕ
≥ min
Dε ∩Br
ϕ
A (x, y)
ϕ
ϕ
ϕ (y) q
|ϕ (y)|p
+
A
(x,
y)
ϕ dy
ϕ q
A (x, y)
εp
dy,
A (x, y)
ϕ p
Dε ∩Br
ε
ϕ
q
dy .
Xie and Ma Advances in Difference Equations (2017) 2017:10
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We conclude that |Dε ∩ Br | = for all ε > and r > , where Br (x) Br is the ball centered
at x of radius r > . Hence, |Dε | = for all ε > . This completes the proof.
Hereafter, we proceed as in the proof of Lemma .. Our goal is also to derive an es˜ = [λ , λ̃] with λ̃ < min{λ ( ),
timate of a priori bounds for (λ, u, v) ∈ C +,+ , where λ ∈
λ ( ), . . . , λ (n )}.
Lemma . Suppose that Aij (i, j = , ) satisfy conditions (C), (C), and (C). Then there
exists a constant l > such that u C () ≤ l and v C () ≤ l whenever (λ, u, v) ∈ C +,+ and
˜
λ ∈ .
Proof Indeed, arguing by contradiction, if this were not true, then there would exist
˜
{(λn , un , vn )}∞
n= ⊂ × H () × H () such that one of the following three cases holds:
˜ as n → ∞, vn ≤ l, and (un , vn ) = G(λn , un , vn );
. un → ∞, λn → λ ∈
˜ as n → ∞, un ≤ l, and (un , vn ) = G(λn , un , vn );
. vn → ∞, λn → λ ∈
˜ as n → ∞, and (un , vn ) = G(λn , un , vn ).
. un → ∞, vn → ∞, λn → λ ∈
We just discuss the case because the other two cases can be dealt with analogously. Let
ϕn =
un
un C ()
ϕn =
,
vn
vn C ()
.
Then it follows that
n
n
∇ϕ ∇ρ dx + (φp,un + φq,vn )ϕn ρ dx = λn ϕ ρ dx,
n
n
∇ϕ ∇ρ dx + (φp,un + φq,vn )wn ρ dx = λn ϕ ρ dx.
(.)
Note that {(ϕn , ϕn )}∞
n= is bounded in H () × H (). Then, without loss of generality, we
suppose that there is ϕ = (ϕ , ϕ ) ∈ H () × H () such that
ϕin → ϕi
as n → ∞ in H (), i = , ,
ϕin → ϕi
as n → ∞ in L (), i = , ,
and ϕin → ϕi (i = , ) as n → ∞ a.e. in .
Using similar arguments as in the proof of Lemma ., we have ϕ = , which implies that
at least one of ϕ and ϕ is not equal to . Without loss of generality, assume that ϕ = .
Then by Lemma . there exists some j ∈ {, , . . . , m} such that ϕ |j = . In view of the
first equation of (.), for any positive ρ ∈ H (j ), we have
j
∇ϕn ∇ρ dx ≤
j
∇ϕn ∇ρ dx +
= λn
j
ϕn ρ dx ≤ λ̃
j
φp,un + φq,v
ϕn ρ dx
n
j
ϕn ρ dx.
(.)
Taking the limit in equation (.) and taking ρ = ψ , where ψ is the first eigenfunction
associated λ (j ), we have
ϕ ψ dx ≤ λ̃
λ (j )
j
which is a contradiction.
ϕ ψ dx,
j
Xie and Ma Advances in Difference Equations (2017) 2017:10
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4 Examples
In this section, we consider system (.) with = (, π) to check the validity of the main
results obtained in Sections and . Notice that λ σn = n (n ∈ N) are the eigenvalues of
the linear eigenvalue problem u + λu = with u() = u(π) = and sin nx is the eigenfunction associated with the eigenvalue n , n ∈ N. In particular, σ = and sin x > on (, π).
In what follows, we consider the following example:
Ai,j (x, y) =
if (x, y) ∈ [(, π ) × ( π , π)] ∪ [( π , π) × (, π )],
otherwise
for all i, j = , . Obviously, Aij (i, j = , ) vanish on the diagonal. It follows from Theorem .
that there exists a positive solution for all λ > . In fact, we see that system (.) with
= (, π) and given Aij (i, j = , ) has a positive solution u = χ sin x and v = χ sin x, where
χ and χ are positive constants satisfying
π
λ = (χ )p
π
sinp x dx + (χ )q
sinq x dx + .
Competing interests
The authors declare that they have no competing interests regarding the publication of this paper.
Authors’ contributions
Both authors, XHX and LM, contributed substantially to this paper, participated in drafting and checking the manuscript,
and have approved the version to be published.
Acknowledgements
Special thanks to the anonymous referees for very useful suggestions. The research has been supported by the Natural
Science Foundation of China (11361004). Xian-Hua Xie is supported by the Bidding Project of Gannan Normal University
(16zb01). Both authors are supported by the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation
Techniques of Gannan Normal University.
Received: 25 November 2016 Accepted: 26 December 2016
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