R E S E A R C H
Open Access
Positive solutions for a class of sublinear
elliptic systems
Ruyun Ma
*, Ruipeng Chen and Yanqiong Lu
*Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, PR China
Abstract
In this paper, we are concerned with the existence of positive solutions of the semilinear elliptic system
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–
u1=
λ
(g11(u1) +g12(u2) +· · ·+g1n(un)), x∈, –
u2=
λ
(g21(u1) +g22(u2) +· · ·+g2n(un)), x∈, · · ·
–
un=
λ
(gn1(u1) +gn2(u2) +· · ·+gnn(un)), x∈, u1(x) =u2(x) =· · ·=un(x) = 0, x∈
∂
,where
λ
> 0 is a parameter,gij: [0,∞)→[0,∞) is a continuous real function for eachi,j= 1, 2,. . .,n. Under some appropriate assumptions, we show that the above system has at least one positive solution in certain interval of
λ
. The proofs of our main results are based upon bifurcation theory.MSC: 34B15; 34B18
Keywords: sublinear elliptic systems; positive solutions; eigenvalues; bifurcation theory
1 Introduction
Letbe a bounded smooth domain inRN (N≥). In this paper, we study the existence of positive solutions of the semilinear elliptic system
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–u=λ(g(u) +g(u) +· · ·+gn(un)), x∈, –u=λ(g(u) +g(u) +· · ·+gn(un)), x∈,
· · ·
–un=λ(gn(u) +gn(u) +· · ·+gnn(un)), x∈, u(x) =u(x) =· · ·=un(x) = , x∈∂,
(.)
whereλ> is a bifurcation parameter,gij: [,∞)→[,∞) is a continuous real function for eachi,j= , , . . . ,n.
A solution of (.) is a pair (λ,U) := (λ, (u,u, . . . ,un))∈(,∞)×[C(¯)]n. (λ,U) is called a positive solution of (.) if ui> in for each i= , , . . . ,n. In the follow-ing, (u,u, . . . ,un) also denotes the elements of Rn+ ={(u,u, . . . ,un)∈Rn:ui≥,i= , , . . . ,n}.
The following definitions will be used in the statement of our main results.
Definition .[] Letfi(i= , , . . . ,n) be smooth real functions defined onRn+. Define the
Jacobian of the vector field (f,f, . . . ,fn) as
H(u,u, . . . ,un) = ⎛ ⎜ ⎜ ⎝
∂f
∂u · · ·
∂f
∂un
..
. . .. ...
∂fn ∂u · · ·
∂fn ∂un
⎞ ⎟ ⎟ ⎠=
⎛ ⎜ ⎜ ⎝
f · · · fn ..
. . .. ... fn · · · fnn
⎞ ⎟ ⎟ ⎠.
If ∂fi
∂uj ≥ (i=j) for all (u,u, . . . ,un)∈R
n
+, then the semilinear elliptic system
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–u=f(u,u, . . . ,un), x∈, –u=f(u,u, . . . ,un), x∈,
· · ·
–un=fn(u,u, . . . ,un), x∈, u(x) =· · ·=un(x) = , x∈∂
(.)
is said to be cooperative. Similarly,His called a cooperative matrix.
Definition .[] Ann×nmatrixAis reducible if for some permutation matrixQ,
QAQT=
B
C D
,
whereBandDare square matrices, andQT is the transpose ofQ. Otherwise,Ais irre-ducible.
In the past few years, the existence of positive solutions to sublinear semilinear elliptic systems with two equations have been extensively studied, see for example, [–] and the references therein. The sublinear condition plays an important role. Very recently, Wu and Cui [] considered the existence, uniqueness and stability of positive solutions to the sublinear elliptic system (.). By using bifurcation theory and the continuation method, they proved the following.
Theorem A Assume that
(H) Eachgij(i,j= , , . . . ,n)is a smooth real function defined onR+satisfying
gij()≥.
(H) gij(s)≥,(gij(s)/s)≤for alls≥.
(H) lims→∞gij(ss) = .
(i) If at least one ofgij()(i= , , . . . ,n)is positive and matrixG= (gij())n×nis irreducible,then(.)has a unique positive solutionU(λ) = (u(λ),u(λ), . . . ,un(λ)) for allλ> ;
(ii) Ifgij() = ,gij() > for eachi,j= , , . . . ,nand matrixG= (gij())n×nis
irreducible,then for someλ∗> , (.)has no positive solution whenλ≤λ∗,and(.)
has a unique positive solutionU(λ) = (u(λ),u(λ), . . . ,un(λ))forλ>λ∗.
We are interested in the existence of positive solutions of (.) under weaker assump-tions. More concretely, we consider the existence of positive solutions of (.) from the following two aspects: (a) To obtain the counterpart of Theorem A(ii) under the weaker assumptions than those of []. In other words, we will not assume thatgij(i,j= , , . . . ,n) are smooth functions any more. (b) Furthermore, we will also consider the case that
lims→
gij(s)
s (i,j= , , . . . ,n) may not exist. More precisely, the following two theorems, which are the main results of the present paper, shall be proved.
Theorem . Suppose that
(A) gij: [,∞)→[,∞)(i,j= , , . . . ,n)are continuous real functions satisfying
gij() = ; gij(s) > , s> .
(A) There exist constantsg
ij∈(,∞)such that
gij=lim
s→
gij(s)
s , ∀i,j= , , . . . ,n.
(A) lims→∞gijs(s)= ,∀i,j= , , . . . ,n.
If the matrix(gij)n×nis irreducible,then there existsλˆ> such that(.)has no positive solution forλ<λˆand(.)has at least one positive solution forλ≥ ˆλ.
Theorem . Let(A)and(A)hold.Assume the following.
(A) For eachi,j∈ {, , . . . ,n},there existg
ij,gij∈(,∞)such that
g
ij=lim infs→
gij(s)
s ≤lim sups→
gij(s) s =gij.
(A) The matrixJα:= (( –α)g
ij+αgij)n×nis irreducible,whereα∈[, ].
Then for someλ˜> , (.)has at least one positive solution forλ>λ˜.
Remark . It follows from (A) and (A)that the matrices (gij)n×nandJα,α∈[, ] are
all cooperative.
Remark . We note that our assumptions in Theorems . and . are weaker than those
of Theorem A, and, accordingly, our results are weaker than Theorem A. Since we just suppose thatgijis continuous, we can only obtain the continua of positive solutions of (.) by applying bifurcation techniques, which are not necessarily curves of positive so-lutions, and thus the uniqueness and stability of positive solutions are not investigated. In [], the authors obtained a smooth curve consisting of positive solutions of (.) by assum-ing stronger assumptions, under which the uniqueness and stability of positive solutions can be achieved.
The rest of the paper is arranged as follows. In Section , we recall some basic knowl-edges on the maximum principle of cooperative systems as well as the eigenvalues of co-operative matrices. Finally in Section , we prove our main results Theorems . and . by applying bifurcation theory.
2 Preliminaries
We shall essentially work in Banach spaceX= [C(¯)]n, here
C(¯) =
u∈C(¯) :u(x) = ,x∈∂.
The norm ofU∈Xwill be defined asUX=nl=ul, where · denotes the norm of C(¯). We useW,p() andW,p
loc() for the standard Sobolev space. We useN(L) andR(L)
to denote the null and the range space of a linear operatorL, respectively.
Let (λ,U) = (λ, (u,u, . . . ,un)) be a solution of (.). Suppose thatgij: [,∞)→[,∞) (i,j= , , . . . ,n) are smooth real functions. Then we can deduce the eigenvalue problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ξ+λg ξ+λgξ+· · ·+λgnξn= –μξ, x∈,
ξ+λg ξ+λgξ+· · ·+λgnξn= –μξ, x∈, · · ·
ξn+λgnξ+λgnξ+· · ·+λgnn ξn= –μξn, x∈,
ξ(x) =ξ(x) =· · ·=ξn(x) = , x∈∂,
(.)
which can be rewritten as
Lu=Hu+μu, (.)
where
u= ⎛ ⎜ ⎜ ⎝
ξ
.. . ξn
⎞ ⎟ ⎟
⎠, Lu=
⎛ ⎜ ⎜ ⎝
–ξ
.. . –ξn
⎞ ⎟ ⎟
⎠, H=λ
⎛ ⎜ ⎜ ⎝
g · · · gn ..
. . .. ... gn · · · gnn
⎞ ⎟ ⎟
⎠. (.)
Lemma .[, ] Let Y = [Wloc,p()∩C(¯)]nand Z= [Lp()]nwith p>N.Suppose that
L,H are given as in(.),and H is cooperative and irreducible.Then we have the following:
(a) μ=inf{Re(μ) :μ∈spt(L–H)}is a real eigenvalue ofL–H,wherespt(L–H)is the
spectrum ofL–H.
(b) Forμ=μ,there exists a unique(up a constant multiple)eigenfunctionu∈Y,and
u> in.
(c) Forμ<μ,the equationLu=Hu+μu+fis uniquely solvable for anyf∈Z,and u> as long asf≥.
(d) (Maximum principle)Forμ<μ,assume thatu∈[Wloc,p()∩C(¯)]nsatisfies
Lu≥Hu+μuin,u≥on∂,thenu≥in.
(e) If there existsu∈[Wloc,p()∩C(¯)]nsatisfyingLu≥Huin,u≥on∂,and eitheru≡on∂orLu≡Huin,thenμ> .
L–H, and|μi–μ| → ∞asi→ ∞. We notice thatμi(i≥) are not necessarily real-valued.
In this section, we also need to consider the adjoint operator ofL–H. Let the transpose matrix ofHbe
HT= ⎛ ⎜ ⎜ ⎝
g · · · gn ..
. . .. ... gn · · · gnn
⎞ ⎟ ⎟ ⎠.
Then it is clear that the results in Lemma . are also true for the eigenvalue problem
Lu∗=HTu∗+μu∗,
which is equivalent to
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ξ∗+λgξ∗+λg ξ∗+· · ·+λgnξn∗= –μξ∗, x∈, ξ∗+λg ξ∗+λg ξ∗+· · ·+λgnξn∗= –μξ∗, x∈,
· · ·
ξn∗+λgnξ∗+λgnξ∗+· · ·+λgnn ξn∗= –μξn∗, x∈, ξ∗(x) =ξ∗(x) =· · ·=ξn∗(x) = , x∈∂,
(.)
whereu∗= (ξ∗,ξ∗, . . . ,ξn∗)T. It is easy to verify thatL–HTis the adjoint operator ofL–H, while both are considered as operators defined on subspaces of [L()]n.
The following lemmas are crucial in the proof of our main results.
Lemma .[] Let Y,Z,L and H be the same as in Lemma..Then the principal eigen-valueμof L–H is also a real eigenvalue of L–HT,μ=inf{μ∈spt(L–HT)},and for
μ=μ,there exists a unique eigenfunctionu∗ ∈[Wloc,()∩C(¯)]nof L–HT (up a
con-stant multiple),andu∗ > in.
Lemma .[, Theorem ..] Let n×n matrix A be a nonnegative irreducible matrix. Thenρ(A)is a simple eigenvalue of A,associated to a positive eigenvector,whereρ(A) de-notes the spectral radius of A.Moreover,ρ(A) > .
Lemma .[] Let V be a real Banach space.Suppose that
F:R×V→V
is completely continuous and F(λ, ) = for allλ∈R.Let a,b∈R(a<b)such that u= is the isolated solution of the equation
u–F(λ,u) = , u∈V. (.)
Furthermore,assume that
dI–F(a,·),Br(),
=dI–F(b,·),Br(),
where Br()is an isolated neighborhood of trivial solutions.Let
S=(λ,u) : (λ,u)is a solution of(.)and u= ∪[a,b]× {}.
Then there exists a continuum(i.e.,a closed connected set)CofScontaining[a,b]× {}, and either
(i) Cis unbounded inR×V;or
(ii) C∩[(R\[a,b])× {}]=∅.
Finally,let(λ,ϕ)be the principal eigen-pair of the linear eigenvalue problem
–ϕ=λϕ, x∈,
ϕ= , x∈∂, (.)
such thatϕ> inandϕ= .
3 Proof of the main results
Proof of Theorem.. We extend eachgijto be a nonnegative continuous function, which is still denoted bygij, defined onRin the following way: ifs< , thengij(s)≡gij().
Let us define
F(λ,U) = ⎛ ⎜ ⎜ ⎜ ⎝
u+λ(g(u) +g(u) +· · ·+gn(un))
u+λ(g(u) +g(u) +· · ·+gn(un))
· · ·
un+λ(gn(u) +gn(u) +· · ·+gnn(un)) ⎞ ⎟ ⎟ ⎟
⎠, (.)
whereλ∈R. Then it follows from (A) thatF:R×X→Xis continuous, and (λ,U) = (λ, (, , . . . , )) is always a solution of (.). Moreover, (A) implies thatFis differentiable at (λ,U) = (λ, (, , . . . , )), and
FU
λ, (, , . . . , ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
φ
φ
.. . φn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠=
⎛ ⎜ ⎜ ⎜ ⎝
φ+λ(gφ+gφ+· · ·+gnφn)
φ+λ(gφ+gφ+· · ·+gnφn)
· · ·
φn+λ(gnφ+gnφ+· · ·+gnnφn) ⎞ ⎟ ⎟ ⎟ ⎠
= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
φ
φ
.. .
φn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+λ
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
g
g · · · gn g
g · · · gn ..
. ... . .. ... gn gn · · · gnn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
φ
φ
.. . φn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
φ
φ
.. .
φn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+λJ
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
φ
φ
.. . φn
⎞ ⎟ ⎟ ⎟ ⎟
⎠, (.)
(k,k, . . . ,kn)Tsatisfyingki> (i= , , . . . ,n). Moreover, it is not difficult to verify that ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
–(kϕ)
–(kϕ)
.. . –(knϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠=λ∗J
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
kϕ
kϕ
.. . knϕ
⎞ ⎟ ⎟ ⎟ ⎟ ⎠, (.)
whereλ∗=χλJ. This implies that (kϕ,kϕ, . . . ,knϕ)
Tis a positive eigenvector of the
op-eratorFU(λ∗, (, , . . . , )). Similarly,JThas the same principal eigenvalueχJand the cor-responding eigenvector is (k∗, . . . ,k∗n)T, wherek∗
i (i= , , . . . ,n) is a positive constant. Ob-viously, ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
–(k∗ϕ)
–(k∗ϕ)
.. . –(kn∗ϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠=λ∗J
T ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
k∗ϕ
k∗ϕ
.. . kn∗ϕ
⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
Hence whenλ=λ∗=λχJ,FU(λ∗, (, , . . . , )) is not invertible andλ=λ∗is a potential
bi-furcation point. More precisely, the null space
NFU
λ∗, (, , . . . , )=span(kϕ,kϕ, . . . ,knϕ)T
is one dimensional. In addition, it is easy to see thatFλ(λ,U) andFλU(λ, (, , . . . , )) exist for (λ,U)∈R×X.
We divide the rest of the proof into two steps.
Step . We show that (λ∗, (, , . . . , )) is actually a bifurcation point.
Indeed, the proof of this is similar to the proof of Theorem A(ii), we state it here for the readers’ convenience.
Suppose (h,h, . . . ,hn)T∈R(FU(λ∗, (, , . . . , ))). Then there exists (ψ,ψ, . . . ,ψn)∈X such that
FU
λ∗, (, , . . . , ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ .. . ψn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ .. . ψn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠+λ∗J
⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ .. . ψn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ h h .. . hn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. (.)
Let us consider the adjoint eigenvalue equation ⎛
⎜ ⎜ ⎝
w∗ .. . w∗n
⎞ ⎟ ⎟ ⎠+λ∗JT
⎛ ⎜ ⎜ ⎝
w∗ .. . w∗n
⎞ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎝
w∗ .. . w∗n
⎞ ⎟ ⎟ ⎠+λ∗
⎛ ⎜ ⎜ ⎝
g
· · · gn
..
. . .. ... gn · · · gnn
⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝
w∗ .. . w∗n
⎞ ⎟ ⎟
⎠= , (.)
wherew∗i =k∗iϕ,i= , , . . . ,n. Multiplying the system (.) by (w∗,w∗, . . . ,w∗n)T, multiply-ing the system (.) by (ψ,ψ, . . . ,ψn)T, integrating onand subtracting, then we obtain
n
i=
hiw∗idx=
n
i=
Thus (h,h, . . . ,hn)T∈R(FU(λ∗, (, , . . . , ))) if and only if (.) holds, which implies that
R(FU(λ∗, (, , . . . , ))) is one dimensional.
Next, we verify that
FλU
λ∗, (, , . . . , )
⎛ ⎜ ⎜ ⎝
kϕ
.. . knϕ
⎞ ⎟ ⎟ ⎠∈/RFU
λ∗, (, , . . . , ). (.)
Otherwise, we have
FλU
λ∗, (, , . . . , ) ⎛ ⎜ ⎜ ⎝
kϕ
.. . knϕ
⎞ ⎟ ⎟ ⎠∈RFU
λ∗, (, , . . . , ). (.)
Since
FλU
λ∗, (, , . . . , ) ⎛ ⎜ ⎜ ⎝
kϕ
.. . knϕ
⎞ ⎟ ⎟ ⎠=J
⎛ ⎜ ⎜ ⎝ k
.. . kn
⎞ ⎟ ⎟ ⎠ϕ=χJ
⎛ ⎜ ⎜ ⎝ k
.. . kn
⎞ ⎟ ⎟
⎠ϕ, (.)
multiplying the system (.) by (k∗ϕ,k∗ϕ, . . . ,k∗nϕ)Tand using (.), we can get a
contra-diction that
=χJ·
kk∗+· · ·+knkn∗
ϕdx> .
By using [, Theorem .], we conclude that (λ∗, (, , . . . , )) is a bifurcation point. Fur-thermore, by the Rabinowitz global bifurcation theorem [], there exists a continuumC+
of positive solutions of (.), which joins (λ∗, (, , . . . , )) to infinity inR×X. Clearly,
C+∩{} ×X=∅, (.)
since (.) has only the trivial solution (, (, , . . . , )) whenλ= . Step : We claim thatC+
cannot blow up at some finiteλ∗∈(,∞).
Otherwise, a sequence{(λk,Uk)} ⊂C+
can be taken such that
lim
k→∞λ
k=λ∗, lim
k→∞
UkX=∞, (.)
whereUk= (uk
,uk, . . . ,ukn). LetK:C(¯)→C(¯) be the Green operator of –subject
to Dirichlet boundary conditions,i.e.,u=Kvif and only if
–u=v, x∈,
By the elliptic regularity, (λk,Uk) satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ uk
(x) =λkK[g(uk(x)) +g(uk(x)) +· · ·+gn(ukn(x))], x∈, uk
(x) =λkK[g(uk(x)) +g(uk(x)) +· · ·+gn(ukn(x))], x∈,
· · ·
uk
n(x) =λkk[gn(uk(x)) +gn(uk(x)) +· · ·+gnn(ukn(x))], x∈.
(.)
Heregijalso denotes the Nemytski operator generated by itself. Clearly, (.) is equivalent to ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
uk uk
.. . ukn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠=λ
kK ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
g(uk) +g(uk) +· · ·+gn(ukn) g(uk) +g(uk) +· · ·+gn(ukn)
.. .
gn(uk) +gn(uk) +· · ·+gnn(ukn) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, (.)
whereK=diag(K,K, . . . ,K). It is well known thatK:C(¯)→C(¯) is continuous and
compact, and soKis continuous and compact on (,∞)×X.
Letwki = u
k i
Uk
X (i= , , . . . ,n). Then w
k
i > inand(wk,wk, . . . ,wkn)X= . Dividing both sides of (.) withUk
X, we have
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
wk wk .. . wk n ⎞ ⎟ ⎟ ⎟ ⎟ ⎠=λ
kK ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
g(uk)
Uk X +
g(uk)
Uk
X +· · ·+
gn(ukn)
Uk X
g(uk)
Uk X +
g(uk)
Uk
X +· · ·+
gn(ukn)
Uk X
.. . gn(uk)
Uk X +
gn(uk)
Uk
X +· · ·+
gnn(ukn)
Uk X ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (.)
For eachi,j= , , . . . ,n, from (A) and (A) it follows thatgij(ss)is bounded in [,∞). More-over, we have
gij(ukj)
Uk X≤
gij(ukj)
ukj = gij(ukj)
ukj · ukj
ukj. (.)
Therefore, ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
g(uk)
Uk X+
g(uk)
Uk
X+· · ·+
gn(ukn)
Uk X
g(uk)
Uk X +
g(uk)
Uk
X +· · ·+
gn(ukn)
Uk X
.. . gn(uk)
Uk X +
gn(uk)
Uk
X +· · ·+
gnn(ukn)
Uk X ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
is bounded inX. This together with the compactness ofKimplies that{(wk
,wk, . . . ,wkn)} has a subsequence, denoted by itself, satisfying, inX,
Obviously,w˜j≥ (j= , , . . . ,n) inand(w˜,w˜, . . . ,w˜n)X= . In addition, we have
λ∗> . Or else, letk→ ∞, then by (.) we getw˜j≡ (j= , , . . . ,n) in, which contra-dicts(w˜,w˜, . . . ,w˜n)X= .
We define
+j =x∈:w˜j(x) >
, j =\+j, j= , , . . . ,n. (.)
Then for eachj= , , . . . ,n,
ukj(x) =wkj(x)·UkX→+∞ ask→ ∞,x∈+j,
by Lebesgue control convergence theorem, we get
gij(ukj(x))
ukj(x) → ask→ ∞,x∈
+
j,∀i,j= , , . . . ,n,
which together with (.) yields
gij(ukj(x))
Uk X
→ ask→ ∞,x∈+j,∀i,j= , , . . . ,n. (.)
On the other hand, we know from (A) and (.) that
gij(ukj(x))
Uk X
→ ask→ ∞,x∈j,∀i,j= , , . . . ,n. (.)
Hence we conclude from (.) and (.) that
gij(ukj(x))
Uk X
→ ask→ ∞,x∈,∀i,j= , , . . . ,n. (.)
Now, letk→ ∞in (.), using (.) and the fact thatλ∗> we can obtain
˜
wj(x) = in, for allj= , , . . . ,n,
which contradicts(w˜,w˜, . . . ,w˜n)X= .
Finally, by (.), the connectness ofC+ and above arguments, we can find someλˆ> such that (.) has no positive solution forλ<λˆ, and (.) has at least one positive solution
forλ≥ ˆλ.
To prove Theorem ., we need the following lemmas as required.
By Remark . and Lemma ., the matricesJ= (gij)n×nandJ= (gij)n×nhave the prin-cipal eigenvaluesχ:=ρ(J) > andχ:=ρ(J) > , respectively, and the corresponding
positive eigenvectors are (k,k, . . . ,k
n)T and (k,k, . . . ,kn)T. Moreover, it is easy to ob-tain
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
(k ϕ)
(kϕ)
.. . (k
nϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+
λ
χ
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ g
g · · · gn g
g · · · gn ..
. ... . .. ... g
n gn · · · gnn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
k ϕ
kϕ
.. . k
nϕ
⎞ ⎟ ⎟ ⎟ ⎟
and ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
(kϕ)
(k ϕ)
.. . (k
nϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+ λ χ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
g g · · · gn g g · · · gn
..
. ... . .. ... gn gn · · · gnn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
kϕ
k ϕ
.. . k
nϕ
⎞ ⎟ ⎟ ⎟ ⎟ ⎠= , (.)
where λ is given as in (.). By Lemma ., the matrices(J)T and (J)T have
princi-pal eigenvaluesχandχ, respectively, the associated positive eigenvectors are (k∗,k∗,
. . . ,kn∗)Tand (k∗,k∗, . . . ,kn∗)T. We can easily verify that
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
(k∗ ϕ)
(k∗ϕ)
.. . (k∗
n ϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+ λ χ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ g
g · · · gn
g
g · · · gn
..
. ... . .. ... g
n gn · · · gnn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
k∗ ϕ
k∗ϕ
.. . k∗
n ϕ
⎞ ⎟ ⎟ ⎟ ⎟ ⎠= (.) and ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
(k∗
ϕ)
(k∗ϕ)
.. . (k∗
nϕ)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠+ λ χ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
g g · · · gn g g · · · gn
..
. ... . .. ... gn gn · · · gnn
⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
k∗
ϕ
k∗ϕ
.. . k∗
nϕ
⎞ ⎟ ⎟ ⎟ ⎟ ⎠= . (.)
Let⊂(,∞)×Xbe the closure of the set of positive solutions to (.). We extend each gijto be a function defined onRby
˜
gij(s) =
gij(s), s∈[,∞), gij(), s∈(–∞, ),
(.)
theng˜ij(s)≥ onR. Let (λ,U)∈(,∞)×Xbe a solution of
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–u=λ(g˜(u) +g˜(u) +· · ·+g˜n(un)), x∈, –u=λ(g˜(u) +g˜(u) +· · ·+g˜n(un)), x∈,
· · ·
–un=λ(g˜n(u) +g˜n(u) +· · ·+g˜nn(un)), x∈, u(x) =u(x) =· · ·=un(x) = , x∈∂.
(.)
Then by (.), for eachi= , , . . . ,n,
ui(x) =λK
˜
gi
u(x)
+g˜i
u(x)
+· · ·+g˜in
un(x)
≥, x∈,
principle of elliptic boundary value problems, we have
=ui=λK
˜
gi(u) +g˜i(u) +· · ·+g˜in(un)
> , x∈,
which is a contradiction. Therefore, the closure of the set of nontrivial solutions of (.) is exactly.
In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping
λ:X→X,
λ(U) =U–λKG(U), (.)
whereKis given as in (.), and
G(U)(x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
˜
g(u)(x) +g˜(u)(x) +· · ·+g˜n(un)(x)
˜
g(u)(x) +g˜(u)(x) +· · ·+g˜n(un)(x) ..
.
˜
gn(u)(x) +g˜n(u)(x) +· · ·+g˜nn(un)(x) ⎞ ⎟ ⎟ ⎟ ⎟
⎠, x∈
is the associated Nemytski operator. ForR> , letBR={U∈X:UX<R}, letd(λ,BR, ) denote the degree ofλonBRwith respect to .
Lemma . Let⊂R+be a compact interval with∩[λ
χ,
λ
χ] =∅.Then there exists a δ> such that
λ(U)= , ∀U∈X, <UX≤δ, ∀λ∈.
Proof Suppose on the contrary that there exist sequences{λk} ⊂and{Uk} ⊂Xso that
λk
Uk= , ∀k∈N, (.)
λk→λ∗ and UkX→, k→ ∞. (.)
Apparently, (.) is equivalent to
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–uk
=λk(g˜(uk) +g˜(uk) +· · ·+g˜n(ukn)), x∈, –uk
=λk(g˜(uk) +g˜(uk) +· · ·+g˜n(ukn)), x∈,
· · ·
–uk
n=λk(g˜n(uk) +g˜n(uk) +· · ·+g˜nn(ukn)), x∈, uk(x) =uk(x) =· · ·=uk
n(x) = , x∈∂
(.)
anduki ≥ (i= , , . . . ,n) in, and thereforeg˜ij(ukj) =gij(ukj),∀i,j= , , . . . ,n. Furthermore, it follows from (A)that, fors> sufficiently small,g
ijs≤gij(s)≤gijs. This together with (.) implies that, forklarge enough,
g ij·u
k j(s)≤gij
Multiplying (.) by (k∗ϕ,k∗ϕ, . . . ,kn∗ϕ)T, multiplying (.) by (uk,uk, . . . ,ukn)T, in-tegrating onand adding, using (.) and the factg˜ij(ukj) =gij(ukj),∀i,j= , , . . . ,n, we know that, forklarge enough,
λ
χ
n
i=
n
j=
g ijk
∗
i ukjϕ
dx=λk
n
i=
n
j=
gij
ukjki∗ϕ
dx
≥λk
n
i=
n
j=
g ijk
∗
i ukjϕ
dx, (.)
and soλk≤ λ
χ forksufficiently large. Similarly, by (.) and (.), we can deduce that λk≥ λ
χforklarge enough. Consequently, forksufficiently large we getλ k∈[λ
χ,
λ
χ], which
contradicts{λk} ⊂.
Corollary . Forλ∈(,λ
χ)andδ∈(,δ),d(λ,Bδ, ) = .
Proof Lemma ., applied to the interval= [,λ], guarantees the existence ofδ> such
that, forδ∈(,δ),
U–τ λKG(U)= , U∈X, <UX≤δ,τ∈[, ].
Hence for anyδ∈(,δ),
d(λ,Bδ, ) =d(I,Bδ, ) = .
Lemma . Suppose thatλ>λ
χ.Then there existsδ> such that
λ(U)=τ ϕ, ∀U∈X: <UX≤δ,∀τ ≥,
whereϕ= (k
ϕ,kϕ, . . . ,knϕ)T.
Proof Suppose on the contrary that there existτk≥ and a sequence{Uk}withUkX> andUk
X→ such that
λ
Uk=τkϕ, ∀k∈N,
which is ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–uk
=λ(g(uk) +g(uk) +· · ·+gn(unk)) +λτkkϕ, x∈,
–uk=λ(g(uk) +g(uk) +· · ·+gn(ukn)) +λτkkϕ, x∈, · · ·
–uk
n=λ(gn(uk) +gn(uk) +· · ·+gnn(ukn)) +λτkknϕ, x∈,
uk
(x) =uk(x) =· · ·=ukn(x) = , x∈∂.
(.)
Clearly, uk
know that, forklarge enough,
λ
χ
n
i=
n
j=
g ijk
∗
i ukjϕ
dx=λ
n
i=
n
j=
gij
ukjki∗ϕ
dx+λτk n
i=
kiki∗ϕ
≥λ
n
i=
n
j=
gij
ukjki∗ϕ
dx
≥λ
n
i=
n
j=
g ijk
∗
i ukjϕ
dx.
Henceλ≤ λ
χ forksufficiently large, which contradictsλ>
λ
χ.
Corollary . Forλ> λ
χ andδ∈(,δ),d(λ,Bδ, ) = .
Proof Let <δ≤δ, whereδis the constant given as in Lemma .. Sinceλis bounded
inB¯δ, there exists a constantC> such thatλ(U)=Cϕ,∀U∈ ¯Bδ. By Lemma ., we get
λ(U)=τCϕ, U∈∂Bδ,τ∈[, ].
Hence,
d(λ,Bδ, ) =d(λ–Cϕ,Bδ, ) = .
Proof of Theorem.. For n∈Nsuch that λ
χ –
n > , letan=
λ
χ –
n, bn=
λ
χ +
n and
ˆ
δ=min{δ,δ}. For anyδ∈(,δˆ), it is easy to see that the assumptions of Lemma . are all
satisfied. Therefore there exists a continuumCof solutions of (.) containing [an,bn]×
{}, and either
(i) Cis unbounded inR×X; or (ii) C∩[(R\[an,bn])× {}]=∅.
By Lemma ., the case (ii) cannot occur, and henceC is unbounded bifurcated from [an,bn]×{}. Note that (.) has only trivial solutions whenλ= , and thereforeC∩({}× X) =∅. Moreover, from Lemma . it follows that for a closed intervalI⊂[an,bn]\[λχ,
λ
χ], ifU∈ {U∈X: (λ,U)∈,λ∈I}, thenUX→ inXis impossible. ThusCmust be bifur-cated from [λ
χ,
λ
χ]. Finally, applying similar methods to the proof of Step of Theorem ., we can show that
λ: (λ,U)∈C⊃
λ
χ
,∞
.
Consequently, (.) has at least one positive solution forλ>λ˜:= λ
χ.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP(No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
Received: 17 October 2013 Accepted: 9 January 2014 Published:30 Jan 2014
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