This paper discusses bifurcation from interval for the elliptic eigenvalue problems with
nonlinear boundary conditions and studies the behavior of the bifurcation components.

1. Introduction

In recent years, much effort has been devoted to the study of the nonlinear elliptic boundary value problems, in particular, to problems which arise in numerous applications, for example, in physical problems involving the steady state temperature distribution [1, 2], in problems of chemical reactions [1, 3], in the theory of stellar structures [4], and in problems of Riemannian geometry [5]. In particular, letΩbe a bounded domain of Euclidean space ℝN, N≥2, with smooth boundary ∂Ω. The nonlinear elliptic boundary value problem is defined as
(1)-Δu=λa(x)f(u)inΩ,∂u∂n+b(x)g(u)=0on∂Ω,
stimulated by a problem of chemical reactor theory [6], where Δ=∑j=1N(∂2/∂xj2), λ>0 is a parameter, and n is the unit exterior normal to ∂Ω. In this paper, we will be devoted to studying the branches of solutions of the problem (1) which bifurcates from infinity.

The asymptotical linear elliptic eigenvalue problems with nonlinear boundary conditions have been studied in [7–9]. It is worth pointing out that Umezu [9], by using a different approach based on topological degree and global bifurcation techniques [10], discusses bifurcation from infinity for (1) with a(·)≡1. They obtained a unique bifurcation value λ∞ from infinity of (1) and there exists an unbounded, closed, and connected component in (0,∞)×C(Ω-), consisting of positive solutions of (1) and bifurcating from (λ,u)=(λ∞,∞). Moreover, they also proved that all the components bifurcate into the region λ<λ∞ or λ>λ∞ under some proper conditions and f(∞)=limu→∞(f(u)/u), g∞=limu→∞(g(u)/u)∈(0,∞). Note that the asymptotical linear case with linear boundary conditions has been studied in [11] and the references therein.

Of course the natural question is as follows: what would happen if f(∞) does not exist? Obviously the previous results cannot deal with the case liminfs→∞f(s)<limsups→∞f(s).

The purpose of this paper is to show the bifurcation from infinity if f(∞) does not exist and obtain the bifurcation of solutions of (1) from an interval not a point. We will make the following assumptions:

a∈Cθ(Ω-) with a(x)>0 in x∈Ω-; b∈C1+θ(∂Ω) with b≥0 and b≢0 on ∂Ω;

f∈C1([0,∞)) and there exist constants f∞,f∞∈(0,∞) and functions h1,h2∈C1([0,∞)), such that
(2)f∞=liminfs→∞f(s)s,f∞=limsups→+∞f(s)s,f∞s+h1(s)≤f(s)≤f∞s+h2(s),s∈[0,∞),

with
(3)hj(s)=o(|s|)ass⟶∞,j=1,2;

g∈C1([0,∞)) and there exist constants g∞∈(0,∞) and functions k∈C([0,∞)), such that
(4)g(s)=g∞s+k(s),k(s)=o(|s|)ass⟶+∞.

Let X=C(Ω-) be the space of continuous functions on Ω-. Then it is a Banach space with the norm
(5)∥u∥=max{|u(x)|∣x∈Ω-}.

Say a solution u∈C2(Ω-) of (1) is positive if u>0 on Ω.

Definition 1 (see [<xref ref-type="bibr" rid="B12">12</xref>, page 450]).

A solution set 𝒮 of (1) is said to bifurcate from infinity in the interval [a,b], if

the solutions of (1) are, a priori, bounded in X for λ=a and λ=b,

there exists {(λn,un)}⊂𝒮 such that {λn}⊂[a,b] and ∥un∥→∞.

By a constant λ1 we denote the first eigenvalue of the eigenvalue problem:
(6)-Δφ=λa(x)φinΩ,∂φ∂n+b(x)g∞φ=0on∂Ω.
It is well known (cf. Krasnosel’skii [13]) that λ1 is positive and simple and that it is a unique eigenvalue with positive eigenfunctions φ1∈C2+θ(Ω-). In what follows, the positive eigenfunction φ1 is normalized as ∥φ1∥=1.

Theorem 2.

Assume that (A1)–(A3) hold. Then for any σ∈(0,λ1/f∞), [λ1/f∞-σ,λ1/f∞+σ] is a bifurcation interval from infinity of (1), and there exists no bifurcation interval from infinity of (1) in the set (0,∞)∖[λ1/f∞-σ,λ1/f∞+σ]. More precisely, there exists an unbounded, closed, and connected component in (0,∞)×C(Ω-), consisting of positive solutions of (1) and bifurcating from [λ1/f∞-σ,λ1/f∞+σ]×{∞}.

Theorem 3.

Assume that (A1)–(A3) hold. Suppose that
(7)liminfu→∞h1(u)>f∞g∞limsupu→∞k(u),(8)(resp.,limsupu→∞h2(u)<f∞g∞liminfu→∞k(u)).
Then all the components obtained by Theorem 2 bifurcate into the region λ<λ1/f∞ (resp., λ>λ1/f∞).

2. Bifurcation Theorem from Interval for Compact Operator

Our main tools in the proof of Theorems 2-3 are topological arguments and the global bifurcation theorems for mappings which are not necessary smooth.

Let V be a real Banach space. Let F:ℝ×V→V be completely continuous. Let us consider the equation
(9)u=F(λ,u).

Lemma 4 (see [<xref ref-type="bibr" rid="B12">12</xref>, Theorem 1.3.3]).

Let V be a Banach space. Let F:ℝ×V→V be completely continuous, and let a,b∈ℝ(a<b) be such that the solutions of (9) are, a priori, bounded in V for λ=a and λ=b. That is, there exists an R>0 such that
(10)u≠F(a,u),u≠F(b,u),∀u:∥u∥≥R.
Furthermore, assume that
(11)d(I-F(a,·),BR(0),0)≠d(I-F(b,·),BR(0),0)
for R>0 large. Then there exists a closed connected set 𝒞 of solutions of (9) that is unbounded in [a,b]×V, and either

𝒞 is unbounded in λ direction or else

there exists an interval [c,d] such that (a,b)∩(c,d)=∅ and 𝒞 bifurcates from infinity in [c,d]×V.

3. Reduction to a Compact Operator Equation

To establish Theorem 2, we begin with the reduction of (1) to a suitable equation for compact operators. According to Gilbarg and Trudinger [14], let 𝒦:Cθ(Ω-)→C2+θ(Ω-) be the resolvent of the linear boundary value problem:
(12)-Δu=ϕinΩ,∂u∂n+b(x)g∞u=0on∂Ω.
By Amann [15, Theorem 4.2], 𝒦 is uniquely extended to a linear mapping of C(Ω-) compactly into C1(Ω-) and it is strongly positive, meaning that 𝒦ϕ>0 on Ω- for any ϕ∈C(Ω-) with the condition that ϕ≥0 and ϕ≢0 on Ω-.

Let ℛ:C1+θ(∂Ω)→C2+θ(Ω-) be the resolvent of the linear boundary value problem:
(13)-Δu=0inΩ,∂u∂n+b(x)g∞u=ψon∂Ω.
According to Amann [7, Section 4], ℛ is uniquely extended to a linear mapping of C(∂Ω) compactly into C(Ω-). By the standard regularity argument, problem (1) is equivalent to the operator equation:
(14)u=λ𝒦[af(u)]+ℛ[bτ(-k(u))]inC(Ω-).
Here τ:C(Ω-)→C(∂Ω) is the usual trace operator.

Proposition 5.

Let (A1), (A2), and (A3) hold. If [α,β] is a bifurcation interval from infinity of the set of nonnegative solutions of (1), then one has (α,β)⊃[λ1/f∞,λ1/f∞]. Moreover, there exist constants ϵ>0 small and M>0 large such that any nonnegative solution u of (1) is positive on Ω- whenever dist(λ,[λ1/f∞,λ1/f∞])<ϵ and ∥u∥≥M.

Proof.

Let (λj,uj) be nonnegative solutions of (1) with λ=λj such that
(15)∥uj∥⟶∞asj⟶∞,λj∈[α,β].
If
(16)vj=uj∥uj∥,
then we have
(17)vj=λj𝒦[af(uj)∥uj∥]+ℛ[bτ(-k(uj)∥uj∥)]inC(Ω-).
From conditions (3) and (6), for any ϵ>0, there exist constants dϵ,cϵ>0 such that
(18)f(s)-f∞s≤ϵs+dϵ,∀u≥0,(19)f(s)-f∞s≥-ϵs-dϵ,∀u≥0,(20)|k(s)|≤ϵs+cϵ,∀u≥0.
This implies that both f(uj)/∥uj∥ and k(uj)/∥uj∥ are bounded in C(Ω-). By the compactness of 𝒦 and ℛ, it follows from (17) that there exist a function v0∈C(Ω-) and a subsequence of {vj}, still denoted by {vj}, such that
(21)vj⟶v0inC(Ω-)asj⟶∞.
By (15) it follows from (18)–(20) that
(22)limsupj→∞maxx∈Ω-(f(uj)∥uj∥-f∞vj)≤ϵ,liminfj→∞minx∈Ω-(f(uj)∥uj∥-f∞vj)≥-ϵ,limsupj→∞∥k(uj)∥uj∥∥≤ϵ.
Since ϵ is arbitrary, it follows that
(23)limsupj→∞f(uj)∥uj∥≤f∞v0inC(Ω-),liminfj→∞f(uj)∥uj∥≥f∞v0inC(Ω-).
Let λj→λ^ and f(uj)/∥uj∥→w0 as j→∞. Then in view of (17),
(24)v0=λ^𝒦[aw0].

We claim that
(25)λ^∈[λ1f∞,λ1f∞].

Since
(26)f∞v0≤w0≤f∞v0,
it follows from (26) that
(27)λ^𝒦[af∞v0]≤v0≤λ^𝒦[af∞v0],
which implies
(28)λ^f∞λ1≤1≤λ^f∞λ1.
That is,
(29)λ1f∞≤λ^≤λ1f∞.

Since ∥v0∥=1 and v0≥0, the strong positivity of 𝒦 ensures that v0>0 on Ω-, and accordingly, vj>0 on Ω- for j large enough and so is uj from (16). This leads to the latter part of assertions of this proposition. The proof is complete.

4. Existence of Bifurcation Values from Infinity

This section is devoted to the study of the existence of bifurcation values from infinity for (1). To do this, we associate (1) with a nonlinear mapping Φ(λ,u):(0,∞)×C(Ω-)→C(Ω-):
(30)Φ(λ,u):=u-λ𝒦[af(|u|)]+ℛ[bτ(k(|u|))].
We note that a nonnegative u∈C(Ω-) attains (1) if and only if Φ(λ,u)=0.

In this section, we will apply Lemma 4 to show that, for any σ∈(0,λ1/f∞), the interval [λ1/f∞-σ,λ1/f∞+σ] is a bifurcation interval from infinity for (30) and consequently [λ1/f∞-σ,λ1/f∞+σ] is a bifurcation interval from infinity of the nonnegative solutions of (1).

In fact, if [λ1/f∞-σ,λ1/f∞+σ] is a bifurcation interval from infinity for (30), then, according to Definition 1, we have that

the solutions of (30) are, a priori, bounded in X for λ=λ1/f∞-σ and λ=λ1/f∞+σ,

there exists {(λn,un)}⊂𝒮 such that {λn}⊂[λ1/f∞-σ,λ1/f∞+σ] and ∥un∥→∞.

Let {λnj} be any convergent subsequence of {(λn,un)}, and let
(31)limj→∞λnj=λ♯.

We claim that
(32)λ♯∈[λ1f∞,λ1f∞]andunj>0onΩ-,ifjislargeenough.

Indeed, as in the proof of Proposition 5, we have the same conclusion that there exist some w0∈C(Ω-) and a λ♯ such that
(33)v0=λ♯𝒦(a|w0|).
This together with the strong positivity of 𝒦 implies that
(34)v0>0,onΩ-.
Since
(35)f∞v0≤|w0|≤f∞v0,
it follows from (34) that
(36)λ♯𝒦[af∞v0]≤v0≤λ♯𝒦[af∞v0],
which implies(37)λ♯f∞λ1≤1≤λ♯f∞λ1.
That is,
(38)λ1f∞≤λ♯≤λ1f∞.

From (34), it follows that vj>0 on Ω- for j large enough and so is uj from (16). Therefore, [λ1/f∞-σ,λ1/f∞+σ] is actually an interval of bifurcation from infinity for (1).

To prove that [λ1/f∞-σ,λ1/f∞+σ] is a bifurcation interval from infinity for (30), two lemmas on the nonexistence of solutions will be first shown.

LetΦp:(0,∞)×C(Ω-)→C(Ω-)be defined as
(39)Φp(λ,u):=u-λ𝒦[af(|u|)]+p(λ)ℛ[bτ(k(|u|))].
Here p:[0,λ1/f∞]→[0,1] is a smooth cut-off function such that
(40)p(λ)={0nearλ=0,1nearλ=λ1f∞.

Lemma 6.

Let Λ⊂ℝ+ be a compact interval with Λ∩[λ1/f∞,λ1/f∞]=∅. If (A1)–(A3) hold, then there exists a constant r>0 such that
(41)Φp(μ,u)≠0,μ∈Λ,u∈C(Ω-):∥u∥≥r.

Proof.

Assume on the contrary that there exist μj≥0, uj∈C(Ω-), and μ0∈Λ such that
(42)Φp(μj,uj)=0,μj⟶μ0,∥uj∥⟶∞,asj⟶∞.
The same argument as that in the proof of Proposition 5 gives a contradiction that λ1/f∞≤μ0≤λ1/f∞. This is a contradiction. The proof of Lemma 6 is complete.

Lemma 7.

Let any λ>λ1/f∞ be fixed. Assume that (A1)–(A3) hold. Then there exists a constant r>0 such that
(43)Φ(λ,u)≠tφ1,t∈[0,1],u∈C(Ω-):∥u∥≥r.

Proof.

Assume on the contrary that there exist λ0∈(λ1/f∞,∞), t0,tj∈[0,1], and uj∈C(Ω-) which can be taken such that
(44)Φ(λ0,uj)=tjφ1,tj⟶t0,∥uj∥⟶∞,asj⟶∞.
Using the same argument as that in the proof of Proposition 5, we can obtain a subsequence of {uj}, still denoted by {uj}, which may satisfy that uj>0 on Ω- for all j≥1. It follows that
(45)uj=λ0𝒦[af(uj)]-ℛ[bτ(k(uj))]+tjφ1,tj⟶t0∈[0,1],∥uj∥⟶∞asj⟶∞.

For a function φ we let 〈φ〉={sφ:s∈ℝ}. The projection theorem derives the orthogonal decomposition of the Lebesgue space L2(Ω) as
(46)L2(Ω)=〈φ1〉⊕W;u=sφ1+w.
Here the eigenfunction φ1 satisfies ∥φ1∥L2(Ω)=1 within the proof of this lemma, W is the orthogonal complement of 〈φ1〉 in L2(Ω), and s=∫Ωuφ1dx. It follows that the orthogonal decomposition of uj is given as
(47)uj=sjφ1+wj,sj=∫Ωujφ1dx>0,wj∈W.
By the regularity argument, (45) gives that uj∈C2+θ(Ω-), and thus
(48)-Δuj=λ0af(uj)+tjλ1aφ1inΩ,∂uj∂n+b(x)g∞uj=-b(x)k(uj)on∂Ω.
By Green's formula it follows that
(49)∫Ω((λ0af(uj)+tjλ1aφ1)φ1-λ1aujφ1)dx=∫Ω((-Δuj)φ1+uj(Δφ1))dx=∫∂Ωbk(uj)φ1dσ.
Here dσ is the surface element of ∂Ω. This implies that
(50)∫∂Ωbk(uj)φ1dσ≥(λ0f∞-λ1)∫Ωaujφ1dx+λ0∫Ωah1(uj)φ1dx.
Hence assertion (19) gives
(51)∫∂Ωbk(uj)∥uj∥φ1dσ≥f∞(λ0-λ1+λ0ϵf∞)×∫Ωaujφ1dx∥uj∥-λ0∥a∥dϵ∥φ1∥L1(Ω)∥uj∥.
Now use again for (48) the same procedure as in the proof of Proposition 5; then we see that some subsequence of {uj/∥uj∥}, still denoted by {uj/∥uj∥}, tends to a positive function v0 in C(Ω-). Take ϵ>0 so small that λ0-(λ1+λ0ϵ)/f∞>0. Then combining (51) with (23) leads to a contradiction that
(52)0=limj→∞∫∂Ωbk(uj)∥uj∥φ1dσ≥limj→∞{sj∥uj∥f∞(λ0-λ1+λ0ϵf∞)-λ0∥a∥dϵ∥φ1∥L1(Ω)∥uj∥}=f∞(λ0-λ1+λ0ϵf∞)∫Ωav0φ1dx>0.

Lemma 8.

Let λn-=λ1/f∞-1/n, and let λn+=λ1/f∞+1/n, where n∈ℕ satisfies n≥f∞/λ1. Assume that (A1)–(A3) hold. Then there exists constant rn>0 satisfying that rn→∞ as n→∞, such that for any n large enough
(53)deg(Φ(λn-,·),Brn,0)=1,(54)deg(Φ(λn+,·),Brn,0)=0.

Proof.

First we show assertion (53). From Lemma 6, there exists rn>0 such that rn→∞ as n→∞ satisfying that
(55)Φp(λ,u)≠0,∀λ∈[0,λn-],∀u∈C(Ω-):∥u∥=rn.
Since p(0)=0 and p(λn-)=1 for n large enough from (40), by the homotopy invariance and normalization it follows that for any n large enough
(56)deg(Φ(λn-,·),Brn,0)=deg(Φp(λn-,·),Brn,0)=deg(Φp(0,·),Brn,0)=deg(IC(Ω-),Brn,0)=1.

Next, we show assertion (54). We may derive from Lemma 7 that
(57)Φ(λn+,u)≠tφ1,∀t∈[0,1],∀u∈C(Ω-):∥u∥≥rn.
So for any n large enough, by the homotopy invariance, it follows that
(58)deg(Φ(λn+,·),Brn,0)=deg(Φ(λn+,·)-φ1,Brn,0)=0.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>.

For any fixed n∈ℕ with λ1/f∞-1/n>0, set αn=λ1/f∞-1/n and βn=λ1/f∞+1/n. It is easy to verify that, for any fixed n large enough, there exists rn such that rn→∞ as n→∞ satisfying that, for any r≥rn, it follows from Lemmas 6–8 that all conditions of Lemma 4 are satisfied. So there exists a closed connected component 𝒞n of solutions (14) that is unbounded in [αn,βn]×C(Ω-) and either

𝒞n is unbounded in λ direction or else

there exists an interval [c,d] such that (αn,βn)∩(c,d)=∅ and 𝒞n bifurcates from infinity in [c,d]×C(Ω-).

By Lemma 6, the case (ii) cannot occur. Thus 𝒞n is unbounded bifurcated from [αn,βn]×∞ in ℝ×C(Ω-). Furthermore, we have from Lemma 6 that, for any closed interval I⊂[αn,βn]∖[λ1/f∞,λ1/f∞], if u∈{u∈C(Ω-)∣(λ,u) are solutions of (14), λ∈I}, then ∥u∥→∞ in C(Ω-) is impossible. So 𝒞n must be bifurcated from [λ1/f∞,λ1/f∞]×{∞}.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>.

Under condition (7), assume to the contrary that there exist positive solutions uj of (1) with λ=λj≥λ1/f∞ such that λj→λ1/f∞ and ∥uj∥→∞ as j→∞. If vj=uj/∥uj∥, then the same argument as that in the proof of Proposition 5 shows that the existence of a positive function v0∈C(Ω-) such that a subsequence of {vj}, still denoted by {vj}, tends to v0 in C(Ω-). It follows that for any j large enough we have
(59)vj(x)>minx∈Ω-v0(x)2onΩ-,
which implies that
(60)minx∈Ω-uj⟶∞asj⟶∞.
Now we set
(61)h1*=liminfu→∞h1(u)∈(-∞,∞],k*=limsupu→∞k(u)∈[-∞,∞).
We here consider only the case where h1*∈(-∞,∞) and k*∈(-∞,∞). Either the case h2*=∞ or the case k*=-∞ can be dealt with in a similar way with a minor modification. It follows from (60) that, for any ε>0, there exists j1>1 such that, for any j≥j1,
(62)h1*-ε<h1(uj(x))onΩ-,k(uj(x))<k*+εonΩ-.
From a computation using Green's formula, it follows that, for any j≥j1,
(63)(λ1-λjf∞)∫Ωaujφ1dx≥λj∫Ωah1(uj)φ1dx-∫∂Ωbk(uj)φ1dσ>λ1f∞(h1*-ε)∫Ωaφ1dx-(k*+ε)∫∂Ωbφ1dσ.
As an application of Green's formula, it follows that
(64)∫Ωaφ1dx=1λ1∫∂Ωbg∞φ1dσ.
From these two assertions combined, we obtain that, for any j≥j1,
(65)(λ1-λjf∞)∫Ωaujφ1dx>(λ1(h1*-ε)g∞f∞λ1-(k*+ε))∫∂Ωbφ1dσ.
On the right-hand side, we see from (7) that
(66)limε→0(λ1(h1*-ε)g∞f∞λ1-(k*+ε))=h1*g∞f∞-k*>0.
This means that, for any j large enough,
(67)(λ1-λjf∞)∫Ω-aujφ1dx>0,
which contradicts the assumption λj≥λ1/f∞. So case (7) has been proven, case (8) can be also verified in the same method, and the proof is complete.

Acknowledgments

This research is supported by the NSFC (nos. 11301059 and 71171035), HSSF of Ministry of Education of China (no. 13YJA790078), China, Postdoctoral Science Foundation Funded Project (nos. 201104602 and 20100481239).

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