# Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

Authors: Benlong Xu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This paper mainly dealt with the exact number and global bifurcation of positive solutions for a class of semilinear elliptic equations with asymptotically linear function on a unit ball. As byproducts, some existence and multiplicity results are also
obtained on a general bounded domain.


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## Body

## 1. Introduction

In this paper, we are concerned with positive solutions of the following elliptic equation subject to homogeneous Dirichlet boundary condition(Pλ)-Δu=λf(u),inΩ,u=0,on∂Ω,
where Ω is a smooth bounded domain in RN, λ is a positive parameter, u∈C2(Ω)∩C(Ω¯), and the function f satisfies the following.(F1)
f:[0,+∞)→(0,+∞) is a positive C1 function, and f is strictly convex; that is, f′(t) is strictly increasing in t∈(0,∞).(F2)
f is asymptotically linear, that is,
(1)limt→∞f(t)t=a∈(0,+∞).For the past years, this problem attracted attentions of many authors. It was studied in [1–4] with f being strictly increasing and was studied in [5–7] with a specific function f(u)=(u-b)2+ϵ which is not increasing.The main goal of this paper is to study the exact number and bifurcation structure of the solutions of(Pλ) on a unit ball Ω, with a general asymptotically linear function f. Some results in this paper (see Section 3) can be viewed as an extension and improvement of that in [7], but the argument approach here is very different to that in [7]. As byproducts, we also get some new results which also hold for general domain Ω (see Section 2). The paper is organized as follows. In Section 2, we study the existence and multiplicity of solutions for problem (Pλ) on a general bounded domain, with some new results complementing those existing in the literature. In Section 3, we study the exact number and global bifurcation structure of positive solutions of (Pλ) on a unit ball.


## 2. Multiplicity of Positive Solutions on a General Domain

Throughout this section, we assume thatΩ is a smooth bounded domain in RN, and f satisfies (F1) and (F2). We also note that, by maximum principle, all solutions of (Pλ) are positive on Ω.Before the statement of our main result, we derive some preliminary lemmas. Though some of them may be known, we provide their proofs for reader’s convenience.Lemma 1.
For anyλ∈(0,λ1/a), (Pλ) is solvable.Proof.
Consider the functional(2)Jλ(u)=∫Ω‍(|∇u|22-λF(u))dx,
where F(u)=∫0u‍f(t)dt.
From (F1) and (F2), it is easy to see that(3)f′(t)<a,
so
(4)F(u)≤au22+f(0)u.
Poincàre’s inequality∫Ω‍u2≤(1/λ1)∫Ω‍|∇u|2, and the imbedding theorem of L2(Ω) to L1(Ω) yield
(5)Jλ(u)≥∫Ω‍|∇u|22dx-aλ2∫Ω‍u2dx-λf(0)∫Ω‍udx≥∫Ω‍|∇u|22dx-aλ2λ1∫Ω‍|∇u|2dx-λf(0)∫Ω‍udx≥12(1-aλλ1)∫Ω‍|∇u|2dx-λf(0)C(∫Ω‍|∇u|2)1/2dx,
so Jλ(u)→∞ as ∥u∥H01(Ω)→∞, where ∥u∥H01(Ω)=(∫Ω‍|∇u|2)1/2dx, and then Jλ(u) is coercive and bounded from below. It is also easy to see that Jλ(u) is weakly lower semi-continuous [8, page 446, Theorem 1]. By applying direct variational methods [9, page 4, Theorem 1.2], we can get the desired result; that is, minu∈H01(Ω)Jλ(u) is reached at some point u(λ), and u(λ) is a solution of (Pλ) when λ∈(0,λ1/a).Lemma 2.
For anyλ>λ1/m, (Pλ) has no solution, where m=inft>0(f(t)/t).Proof.
If not, assume thatu is a solution of (Pλ) for some λ>λ1/m. Multiplying (Pλ) by φ1>0, the normalized positive eigenfunction with respect to the first eigenvalue λ1 of -Δ subject to homogenous Dirichlet boundary condition, and then integrating by parts, we get
(6)λ1∫Ω‍uφ1dx=∫Ω‍-Δuφ1dx=λ∫Ω‍f(u)φ1dx>λ1∫Ω‍uφ1dx,
which is a contradiction.We begin by show the following.Lemma 3.
There exists a numberλ1/a≤Λ≤λ1/m, such that (Pλ) has at least a solution for λ<Λ and has no solution for λ>Λ.Proof.
Let(7)Λ={λ:(Pλ)hasasolution}.
By Lemmas 1 and 2, λ1/a≤Λ≤λ1/m. We need just to prove that if (Pμ) has a solution, then (Pλ) also has a solution for all 0<λ<μ. This can be done by a simple argument of sub-sup solution method, since it is easy to see that any solution of (Pμ) is a super solution of (Pλ) and u≡0 a subsolution.
It is easy to see thatu*≡0 is a subsolution of (Pλ), then a standard sub-super solution method’s argument and comparison theorems give the following lemma.Lemma 4.
If(Pλ) is solvable, then one has a minimal solution uλ, that is, for any solution v of (Pλ), uλ≤v. Moreover, uλ is increasing with respect to λ.Lemma 5.
Ifλ∈(0,λ1/a), then the solution of  (Pλ) is unique.Proof.
Suppose thatv1 and v2 are solutions of (Pλ). Let v=v1-v2, then
(8)-Δv=λ[f(v1)-f(v2)],inΩ,v=0,on∂Ω.
By mean value theorem, v satisfies
(9)-Δv=f′(v-)v,
where v- lies between v1 and v2. Multiplying v and integrating, we get
(10)∫Ω‍|∇v|2dx=λ∫Ω‍f′(v-)v2dx≤aλ∫Ω‍v2dx≤aλλ1∫Ω‍|∇v|2dx,
which implies that v≡0. The proof is complete.Lemma 6.
The minimal solutionuλ is stable, that is, λ1(-Δ-λf′(uλ))≥0, where λ1(-Δ-λf′(uλ)) denotes the first eigenvalue of the following problem:
(11)-Δw-λf′(uλ)w=μw,inΩ,w=0,on∂Ω.Proof.
Suppose on the contrary thatλ1(-Δ-λf′(uλ))=μ<0, and w>0 is the corresponding eigenvector. Let vε=uλ-εφ, then by (Pλ) and (11), we have
(12)-Δvε-λf(vλ)=λf(uλ)-λεf′(uλ)φ-λf(uλ-εφ)-μεφ=-μεφ+o(εφ)>0,
when ε is small enough, and hence vε=uλ-εφ is a super solution of problem (Pλ). On the other hand, 0 is a subsolution of (Pλ), and Hopf’s boundary lemma implies that 0<vε for ε>0 small. An application of sub-sup solution method guarantees that there is a solution u- of (Pλ) satisfying 0<u-≤uλ-εφ in Ω, which is a contradiction with the minimality of uλ. The proof is complete.Now we state our main result.Theorem 7.
Suppose thatf satisfies (F1) and (F2), then there exists Λ∈[λ1/a,λ1/m] (where m=inft>0(f(t)/t)) such that problem (Pλ)(i)
has at least one solution forλ∈(0,Λ) and a unique solution for λ∈(0,λ1/a);(ii)
has no solution forλ∈(Λ,+∞);(iii)
(a) ifΛ=λ1/a, then problem (Pλ) has no solution at λ=Λ, and limλ→Λ-0uλ(x)=+∞ for all x∈Ω, where uλ denotes the unique solution of (Pλ) for λ∈(0,Λ) (see Figure 1),(b) ifΛ>λ1/a, then problem (Pλ) has a unique solution for λ∈(0,λ1/a] and λ=Λ, has at least two solutions for λ∈(λ1/a,Λ) (see Figure 2 for a minimal diagram).Figure 1
Diagram forΛ=λ1/a.Figure 2
Minimal diagram forΛ>λ1/a.Proof.
Statement (i) follows from Lemmas3 and 5. Statement (ii) follows from Lemma 3. Now we give the proof of statement (iii).
(a) SupposeΛ=λ1/a. The solution (Pλ) bifurcates at infinity near Λ=λ1/a (see [2, 10] for details). On the other hand, (Pλ) has a unique solution uλ for λ∈(0,λ1/a), and no solution for λ>λ1/a. Therefore the bifurcation curve from infinity is on the left of λ=λ1/a, and hence limλ→Λ-0uλ(x)=+∞ for all x∈Ω by the expression of the bifurcation solution in Theorem 13 in Section 3.
If(PΛ) has a solution, let uΛ denote the minimal solution of (Pλ). By Lemma 4, uλ≤uΛ for λ∈(0,Λ), contradicting limλ→Λ-0∥uλ∥∞=∞.
(b) For clarity, the proof will be divided into 3 steps.
Step
1. The existence and uniqueness of solutions of (Pλ) for λ=λ1/a.
The existence follows directly from Lemma4. Note that f′<a, and the uniqueness can be proved in a similar way as in the proof of Lemma 5.
Step
2. The existence and uniqueness of solutions of (Pλ) for λ=Λ.
By Lemmas3 and 4, (Pλ) has a minimal solution uλ for any λ∈(0,Λ), and uλ is increasing in λ. Let (λn)⊂(λ1/a,Λ) be any sequence such that limn→∞λn=Λ. Firstly we insure that case (uλn) is L2(Ω) bounded. Suppose the contrary that limn→∞∥uλn∥L2(Ω)=∞. Let cn=∥uλn∥L2(Ω) and vλn=uλn/cn, then
(13)-Δvλn=λncnf(cnvλn),inΩ,vλn=0,on∂Ω.
Sincef(cnvλn)/cn is bounded in L2(Ω), it follows from (13) that vλn is bounded in H01(Ω). Then subject to a subsequence, we may suppose that there exits v*, such that
(14)vλn⇀v*weaklyinH01(Ω),vλn⟶v*stronglyinL2(Ω),vλn⟶v*a.e.inΩ.
Then by letting n→∞, we get from (13) in the weak sense that
(15)-Δv*=aΛv*,inΩ,v*=0,on∂Ω,
with ∥v*∥L2(Ω)=1, and v*>0 by strong maximum principle. Hence aΛ=λ1, that is, Λ=λ1/a, a desired contradiction.
Now in a similar way, the boundedness of(uλn) in L2(Ω) implies that (uλn) is bounded in H01(Ω). Then subject to a subsequence, we may suppose that there exits u*, such that
(16)uλn⇀u*weaklyinH01(Ω),uλn⟶u*stronglyinL2(Ω),uλn⟶u*a.e.inΩ.
Then by letting n→∞, we get
(17)-Δu*=Λf(u*),inΩ,u*=0,on∂Ω,
and the existence is proved.
Now we prove the uniqueness. LetuΛ be the minimal solution of (PΛ) and u- a different solution. Then w:=u--uΛ>0 satisfies
(18)-Δv=Λf′(uΛ+θw)w,inΩ,v=0,on∂Ω,
where θ:Ω→ℝ satisfying 0<θ<1. It follows that λ1(-Δ-Λf′(uΛ+θw))=0, where λ1(-Δ-Λf′(uΛ+θw)) denotes the first eigenvalue of the operator -Δ-Λf′(uΛ+θw) subject to the Dirichlet boundary condition, as defined in Lemma 1. Since f′(uΛ)<f′(uΛ+θw) in Ω, we have that λ1(-Δ-Λf′(uΛ))>λ1(-Δ-Λf′(uλ+θw))=0, which implies that the operator -Δ-Λf′(uΛ) is nondegenerate. Then by the Implicit Function Theorem, the solution of (Pλ) forms a cure in a neighborhood of (Λ,uΛ), which is clearly contradicted to the definition of Λ in (7).
Step
3. Prove that (Pλ) has at least two solutions for λ∈(λ1/a,Λ).
Following the argument in [5], we prove it by variational method of Nehari type (see [11]). As we have known (Lemma 5), there exists a minimal solution uλ of (Pλ) when λ∈(λ1/a,Λ). Now we must look for another solution u(>uλ). Assuming that u=v+uλ, with v>0, then v satisfies
(19)-Δv=λ[f(v+uλ)-f(uλ)],inΩ,v=0,on∂Ω.
For convenience, let g(v)=f(v+uλ)-f(uλ) and G(v)=∫0v‍g(t)dt, then we have
(20)-Δv=λg(v),inΩ,v=0,on∂Ω.
Define
(21)Jλ(v)=∫Ω(|∇v|22-λG(v))dx,Iλ(v)=∫Ω‍(|∇v|2-λvg(v))dx,
and the solution manifold
(22)Mλ={v∈H01(Ω):v>0inΩ,Iλ(v)=0}.
Firstly we show thatMλ≠ϕ for any λ∈(λ1/a,Λ). Let φ1 be the first eigenfunction of -Δ in Ω subject to Dirichlet boundary condition and ∫Ω‍φ12dx=1, then
(23)Iλ(tφ1)=λ1t2-λ∫Ω‍tφ1g(tφ1)dx=t2(λ1-λ∫Ω‍φ1g(tφ1)tdx),limt→∞∫Ω‍φ1g(tφ1)tdx=limt→∞∫Ω‍φ12·g(tφ1)tφ1dx=a.
It follows from (23) that
(24)Iλ(tφ1)<0,
for sufficiently large t if λ∈(λ1/a,Λ).
On the other hand, letω1 be the eigenfunction with ∫Ω‍ω12dx=1 of the first eigenvalue μ1 of the following equation:
(25)-Δω1-λf′(uλ)ω1=μ1ω1,inΩ,ω1=0,on∂Ω.
Sinceuλ is the minimal solution, it follows from Lemmas 4 and 6 that μ1>0. Then
(26)Iλ(sω1)=s2∫Ω‍|∇ω1|2dx-λs∫Ω‍ω1g(sω1)dx=s2∫Ω‍|∇ω1|2dx-λs∫Ω‍[f′(uλ)sω12+o(s2)]dx=s2[∫Ω‍(|∇ω1|2-λf′(uλ)ω12)dx+o(1)]=s2(μ1+o(1)).
Hence Iλ(sω1)>0 when s is small enough. Now it is easy to see that Mλ is not empty. In fact, take w*=tφ1 for some large t, and w*=sω for some small s>0, such that
(27)Iλ(w*)<0,Iλ(w*)>0,
respectively. Define a continuous function G on [0,1], namely,
(28)G(ξ)=Iλ(ξw*+(1-ξ)w*).
Then G(0)>0, G(1)<0, and hence there exist ξ0∈(0,1) such that G(ξ0)=0, that is, Iλ(ξ0w*+(1-ξ0)w*)=0, and Mλ≠ϕ, a desired conclusion.
Sincef is convex, g(v) is convex with respect to v>0 such that
(29)g(v)=g(v)-g(0)≤g′(v)v.
Integrating (29) with respect to v from 0 to v, we get
(30)2G(v)≤g(v)v.
Therefore, on Mλ(31)Jλ(v)=λ2∫Ω‍[g(v)v-2G(v)]dx≥0,
that is, Jλ(v) is bounded from below.
And then we obtain a nonminimal positive solution of(Pλ) by using the Nehari variational method. The proof is complete.Remark 8.
The solutions that we get from the above discussion are weak ones, but a standard elliptic regularity argument shows that they are indeed classical solutions.In view of Theorem7, we want to know what conditions ensure that Λ=λ1/a or Λ>λ1/a. Following [4], we consider the function L(t)=at-f(t). It is easy to see that L(t) is strictly increasing, and hence limt→∞L(t)=L∞ exists (may be +∞). Also note that L(0)=-f(0)<0.Theorem 9.
IfL∞≤0, then Λ=λ1/a; if L∞>0, then Λ>λ1/a.Proof.
(i) IfL∞≤0, then f(t)≥at for all t≥0. We prove that (Pλ) has no solution and hence Λ=λ1/a. Suppose the contrary that u is a solution (Pλ) for λ=λ1/a, then
(32)-Δu=λ1af(u)≥λ1u.
Let φ be a positive eigenfunction of the first eigenvalue λ of -Δ on Ω with Dirichlet boundary condition, that is
(33)Δφ+λ1φ=0,inΩ,φ=0,on∂Ω.
Multiplying (32) by φ>0, and integrating by parts, we get
(34)∫Ω‍(f(u)-au)φdx=0,
which yields that f(u)=au, contradicting the fact that f(0)>0.
(ii) IfL∞>0, we prove that Λ>λ1/a.
Let(λ(s),u(s)) be the bifurcation curve as described in Theorem 13 in Section 3, then
(35)Δu(s)+λ(s)f(u(s))=0,inΩ,u(s)=0,on∂Ω.
It follows from (33) and (35) that
(36)λ(s)∫Ω‍f(u(s))φdx=λ1∫Ω‍u(s)φdx=λ1a∫Ω‍au(s)φdx.
By the fact that u(s)(x)=sφ(x)+z(s)(x)→∞ (s→∞) a.e. in Ω, we have
(37)∫Ω‍au(s)φdx-∫Ω‍f(u(s))φdx=∫Ω‍(au(s)-f(u(s)))φdx>0,
for s sufficiently large. It follows from (36) that λ(s)>λ1/a when s is sufficiently large, which means that the bifurcation curve (λ(s),u(s)) from infinity is on the right of λ=λ1/a, and hence Λ>λ1/a by the definition of Λ in (7). The proof is complete.Now we define another function which is also crucial in studying exact multiplicity in the next section. Let(38)K(t)=tf′(t)-f(t),
then K′(t)=tf′′(t)>0 a.e. in (0,+∞), and K(t) is strictly increasing, and K(0)=-f(0)<0. Denote
(39)limt→∞K(t)=K∞∈(-∞,+∞].Theorem 10.
IfK∞≤0, then Λ=λ1/a; if K∞>0, then Λ>λ1/a.Proof.
IfK∞≤0, then (f(t)/t)′=K(t)/t2<0 for all t>0. It follows that f(t)/t is strictly decreasing and hence f(t)/t>a, which implies that L∞≤0.
On the other hand, ifK∞>0, by
(40)L(t)-K(t)=t(a-f′(t))>0,∀t>0,
we get that L∞>0. Then the conclusion follows for Theorem 9.


## 3. Exact Number and Global Bifurcation of Solutions on a Unit Ball

From Theorem7, the exact number of solutions (Pλ) is now clear in the case of Λ=λ1/a; that is, the solution is unique if it exists. On the other hand, it is far from known in general exactly how may solutions of (Pλ) for λ∈(λ1/a,Λ) if Λ>λ1/a. Using the bifurcation approach developed in [12–14], and also the idea and techniques developed in [7], we solve this problem on the unit ball under some conditions.Throughout this section, we suppose thatΩ is the unit ball in RN centered with the origin.The next remarkable results regarding(Pλ) are due to Gidas et al. [15] and Lin and Ni [16].Lemma 11.
(1) Iff is locally Lipschitz continuous in [0,∞), then all positive solutions of (Pλ) are radially symmetric, that is, u(x)=u(r), r=|x|, and satisfies
(41)u′′+n-1ru′+λf(u)=0,r∈(0,1),u′(0)=u(1)=0.
Moreover, u′(r)<0 for all r∈(0,1], and hence u(0)=max0≤r≤1u(r).
(2) Supposef∈C1(R). If u is a positive solution to (Pλ), and w is a solution of the linearized problem (43) (if it exists), then w is also radially symmetric and satisfies
(42)w′′+n-1r+λf′(u)w=0,r∈(0,1),w′(0)=w(1)=0.The next lemma also plays a key role in this section.Lemma 12.
(1) For anyd>0, there is at most one λd>0 such that (Pλ) have a positive solution u(·) with λ=λd and u(0)=d.
(2) LetT={d>0:(Pλ) have a positive solution with  u(0)=d}, then T is open; λ(d)=λd is a well-defined continuous function from T to R+.Lemma12 is well known; see, for example, [13, 17, 18]. A simple proof of the first part of the lemma can be found in [14]. Because of Lemma 12, we call R+×R+={(λ,d):λ>0,d>0} the phase space, {(λ(d),d):d∈T} the bifurcation curve, and the phase space with bifurcation curve the bifurcation diagram.We will also need the following theorem of bifurcation from infinity.Theorem 13 (see [10, 19]).
Supposef∈C1(R). Let limu→∞f(u)/u=a∈(0,∞) and λ∞=λ1/a. Then all positive solutions of (Pλ) near (λ∞,∞) have the form of (λ(s),sφ+z(s)) for s∈(δ,∞) and some δ>0, where φ is a positive eigenfunction of the first eigenvalue λ1 of -Δ on Ω subjected to Dirichlet boundary condition, lims→∞λ(s)=λ∞, and ∥z(s)∥C2,α(B-n)=o(s) as s→∞.To make bifurcation argument work, a crucial thing is the following result.Letu be a solution of problem (Pλ), then u is called a degenerate solution if the corresponding linearized equation
(43)-Δw=λf′(u)w,inΩ,w=0,on∂Ω,
has a nontrivial solution.Now suppose thatf satisfies (F1), (F2). As in the end of Section 2, let
(44)K(t)=tf′(t)-f(t)K∞=limt→∞K(t).
If K∞>0, then there exists a unique real number β>0, such that
(45)K(t)<0fort∈[0,β);K(t)>0fort∈(β,∞);K(β)=0.Lemma 14.
Suppose thatK∞>0. If u is a degenerate solution of (Pλ), then u(0)>β.Proof .
Suppose the contrary thatu(0)≤β, then
(46)K(u)=uf′(u)-f(u)<0,inΩ∖{0}.
Let w be a nontrivial solution of the corresponding linearized equation (43). From (Pλ) and (43), we get
(47)0=∫Ω‍(-Δwu+Δuw)dx=λ∫Ω‍(uf′(u)-f(u))wdx.
It appears from (46) and (47) that w must change sign in Ω.
In view of Lemma11(2), we suppose that |x|=r1 is a maximal zero in (0,1). We may also suppose that w(x)>0, for all r1<|x|<1. Then
(48)∫Ω∖B(r1)(-Δwu+Δuw)dx=λ∫Ω‍(uf′(u)-f(u))wdx<0,
where B(r1) denotes the ball of radius r1 centered with the origin.
On the other hand, using integration by parts, we have(49)∫Ω∖B(r1)‍(-Δwu+Δuw)dx=-∫∂(Ω∖B(r1))‍∂w∂νuds>0.
a contradiction.Theorem 15.
Suppose thatf satisfies (F1)-(F2) with 0<K∞<aβ. If u is a degenerate solution of (Pλ), then any nontrivial solution of the corresponding linearized equation (43) does not change sign in Ω.Proof .
By Lemma14, maxx∈Ω¯u(x)=u(0)>β. In view of Lemma 11, there exists r*∈(0,1), such that u(r*)=β. Let w be a non-trivial solution of the corresponding linearized equation (43), then w(0)≠0.
We assert thatw(r) has no zeroes on [r*,1). Suppose the contrary and let r1 be the largest zero of w on [r*,1). We may suppose that w>0 in (r1,1). Note that u(r)<β for r∈(r1,1), a similar argument as in the proof of Lemma 14 yields a contradiction.
Now we prove thatw(r) has no zeroes on (0,r*). Suppose the contrary and let r0 be the smallest zero of w(r) on (0,r*). We may suppose that w>0 in B(r0). Multiplying (Pλ) by u-β, (43) by w, subtracting, and integrating on B(r0), we get
(50)∫B(r0)‍[-Δw(u-β)+Δuw]dx=λ∫B(r0)‍[(u-β)f′(u)-f(u)]wdx.
LetJ(t)=(t-β)f′(t)-f(t), then J(0)=-f(0)<0, J(∞)=limt→∞J(u)=K∞-aβ<0, and J′(t)=(t-β)f′′(t)>0 for t>β. Hence J(u)=(u-β)f′(u)-f(u)<0 for x∈B(r0). Then
(51)∫B(r0)‍[(u-β)f′(u)-f(u)]wdx<0.
On the other hand, by Green formula,(52)∫B(r0)‍[-Δw(u-β)+Δuw]dx=-∫∂(B(r0))‍∂w∂ν(u-β)dx>0.
A contradiction occurs from (50), (51), and (52). Hence w(r) has no zeroes in (0,1), that is to say, w does not change sign in Ω. The proof is complete.Now defineF:C02,α(Ω¯)→Cα(Ω¯), by
(53)Fu=Δu+λf(u),
then the linearized operator (Frechèt derivative) is
(54)Fu(λ,u)w=Δw+λf′(u)w.From the maximum principle, all solutions of(Pλ) are positive on Ω. Moreover, if (λ*,u*) is degenerate solution of (Pλ), then by Theorem 15, the nontrivial solution w of (43) does not change sign in Ω, and hence w can be chosen to be positive. Then by Krein-Rutman’s Theorem, N(Fu(λ*,u*))=span{w}, and it follows from Fredholm alternative theorem that codimR(Fu(λ*,u*))=1. Now we prove that Fλ(λ*,u*)∉R(Fu(λ*,u*)). If it is not the case, then there exists v∈C02,α(Ω¯), such that
(55)Δv+λ*f′(u*)v=f(u*).We also have(56)Δw+λ*f′(u*)w=0.Multiplying (55) by w, (56) by v, subtracting, and integrating, we obtain
(57)∫Ω‍f(u*)wdx=0,
a contradiction. As all the conditions of Crandall-Rabinowitz’s bifurcation theorem [20] are satisfied, the solutions of (Pλ) near the degenerate solution (λ*,u*) form a smooth curve which is expressed in the form
(58)(λ(s),u(s))=(λ*+τ(s),u0+sw+z(s)),
where s→(τ(s),z(s))∈R×Z is a smooth function near s=0 with τ(0)=τ′(0)=0,z(0)=z′(0)=0, where Z is a complement of span{w} in X, and w is the positive solution of (43), which is unique if normalized.Substitutingu and λ by expression (58), then differentiating the equation (Pλ) twice, and evaluating at s=0, we have
(59)Δuss+λf(u)uss+2λ′f′(u)us+λf′′(u)us2+λ′′f(u)=0,Δuss+λ*f′(u)uss+λ*f′′(u)w2+λ′′(0)f(u)=0.
Multiplying (59) by w, (43) by uss, subtracting, and integrating, we obtain
(60)τ′′(0)=-λ*∫Ω‍f′′(u*)w3dx∫Ω‍f(u*)wdx<0.By (60) and the Taylor expansion formula of τ(s) at s=0, we conclude that at any degenerate solution (λ*,u*) of (Pλ), the solution curve turns left, that is to say, there is no any solution (λ,u) on the right near (λ*,u*). This observation is very important to our proof of the following theorem.Theorem 16.
Suppose thatΩ is the unit ball in Rn, f satisfies (F1)-(F2), and 0<K∞<aβ. Then for problem (Pλ), (1)
there exist no solutions forλ>Λ,(2)
there exists exactly one solution forλ∈(0,λ1/a]∪{Λ},(3)
there exist exactly two solutions forλ∈(λ1/a,Λ).
Moreover, the solution set{(λ,u)} of (Pλ) forms a smooth curve in the space R×C(Ω¯), which can be roughly described as in Figure 3.Figure 3
Precise bifurcation diagram on a unit ball.Proof.
By Theorem10, Λ>λ1/a, and Theorem 7 tells us that (Pλ) has a unique solution (Λ,uΛ) for λ=Λ, and Implicit Function Theorem implies that (Λ,uΛ) is a degenerate solution. By Theorem 15, non-trivial solution w of the corresponding linearized equation (43) does not change sign in Ω, and we may suppose that w is positive in Ω. Then Crandall-Rabinowitz’s bifurcation theorem [20] and the discussion prior to this theorem imply that the solutions near (Λ,uΛ) form a smooth curve which turns to the left in the phase space. We may call the part of the smooth solution curve {(λ,u)} with u(0)>uΛ(0) the upper branch, and the rest the lower branch. We denote the upper branch by uλ and the lower branch by uλ.
For the upper branch, as long as(λ,uλ) nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing λ. We still denote the extension by (λ,uλ). This process of continuation towards smaller values of λ will not encounter any other degenerate solutions. This is because, if, say, (λ,uλ) becomes degenerate at λ=λ0, the discussion prior to this theorem implies that all the solutions near (λ0,uλ0) must lie to the left side of it, which is a contradiction. Lemma 12 tells us that λ→uλ(0) is decreasing. So in the progress of extension of (λ,uλ) towards smaller values of λ, there are only the following two possibilities.(i)
The upper branch(λ,uλ) stops at some (0,u0), and u0(0)>uΛ(0).(ii)
∥uλ∥∞ goes to infinity as λ→λ~+0,  0≤λ~<Λ.
But case (i) cannot happen, since(0,u0) is obviously not a solution of  (Pλ). Hence case (ii) happens. We assert that λ~=λ1/a. In fact, let {λn} be an arbitrary sequence such that λn→λ~. Denote Mn=∥un∥∞, vn=un/Mn, then Mn→∞ and
(61)Δvn+λnf(Mnvn)Mn=0,inΩ,v=0,on∂Ω.
Since f(Mnvn)/Mn is bounded, by Sobolev Imbedding Theorems and standard regularity of elliptic equation, it is easy to see that {vn} has a subsequence, still denoted by {vn}, such that vn→v in C2,α(Ω)(n→∞), for some v∈C2,α(Ω), v>0 in Ω. Letting n→∞ in (61), we get
(62)Δv+λ~av=0,inΩ,v=0,on∂Ω,
which implies that λ~=λ1/a.
Now we study the structure of the lower branch. As in the case of upper branch, as long as(λ,uλ) nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing λ. We still denote the extension by (λ,uλ). This process of continuation towards smaller values of λ will not encounter any other degenerate solutions. Lemma 12 implies that λ→uλ(0) is increasing. So in the progress of extension of (λ,uλ) towards smaller values of λ, there are only the following two possibilities.(i)
The lower branch(λ,uλ) stops at some (0,u0) with u0(0)>0.(ii)
The lower branch(λ,uλ) stops at some (λ0,0) with 0≤λ0<Λ.
As before, case (i) will not happen. Then case (ii) happens. Byf(0)>0, it is easy to see that λ0=0. That is to say, the lower branch of solutions extends till the origin (0,0) in the phase plane.
By the above argument, we obtain a smooth positive solution curve which consists of an upper branch{(λ,uλ)} and a lower branch {(λ,uλ)}. The lower branch starts from (Λ,uΛ) and stops at (0,0), and λ→uλ(0) is a strictly increasing function. The upper branch {(λ,uλ)} starts from (Λ,uΛ) and stops at (λ1/a,∞), and λ→uλ(0) is a strictly decreasing function with uλ(0) blowing up as λ→λ1/a+0. By Lemma 12, all solutions of (Pλ) are contained in this smooth solution curve, and the complete bifurcation diagram can be described as in Figure 3. The proof is complete.



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*Source: 510943-2013-04-10.xml*