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

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.

1. Introduction

In this paper, we are concerned with positive solutions of the following elliptic equation subject to homogeneous Dirichlet boundary condition (𝑢)𝑃−Δ𝑢=𝜆𝑓,inΩ,𝑢=0,on𝜕Ω,^𝜆 where Ω is a smooth bounded domain in 𝑅^𝑁, 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, lim_𝑡→∞𝑓(𝑡)𝑡=𝑎∈(0,+∞).(1)

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 𝑃_𝜆 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 𝑃_𝜆 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 𝑃_𝜆 on a unit ball.

2. Multiplicity of Positive Solutions on a General Domain

Throughout this section, we assume that Ω is a smooth bounded domain in 𝑅^𝑁, and 𝑓 satisfies (F1) and (F2). We also note that, by maximum principle, all solutions of 𝑃_𝜆 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 any 𝜆∈(0,𝜆₁/𝑎), 𝑃_𝜆 is solvable.

Proof. Consider the functional 𝐽_𝜆(𝑢)=_Ω|∇𝑢|²2−𝜆𝐹(𝑢)𝑑𝑥,(2) where 𝐹(𝑢)=∫^𝑢₀𝑓(𝑡)𝑑𝑡.
From (F1) and (F2), it is easy to see that 𝑓^′(𝑡)<𝑎,(3) so 𝐹(𝑢)≤𝑎𝑢²2+𝑓(0)𝑢.(4)
Poincàre’s inequality ∫_Ω𝑢²≤(1/𝜆₁)∫_Ω|∇𝑢|², and the imbedding theorem of 𝐿²(Ω) to 𝐿¹(Ω) yield 𝐽_𝜆(𝑢)≥_Ω|∇𝑢|²2𝑑𝑥−𝑎𝜆2_Ω𝑢²𝑑𝑥−𝜆𝑓(0)_Ω≥𝑢𝑑𝑥_Ω|∇𝑢|²2𝑑𝑥−𝑎𝜆2𝜆_1Ω|∇𝑢|²𝑑𝑥−𝜆𝑓(0)_Ω≥1𝑢𝑑𝑥21−𝑎𝜆𝜆_1Ω|∇𝑢|²𝑑𝑥−𝜆𝑓(0)𝐶_Ω|∇𝑢|^21/2𝑑𝑥,(5) so 𝐽_𝜆(𝑢)→∞ as ‖𝑢‖_𝐻¹_0(Ω)→∞, where ‖𝑢‖_𝐻¹_0(Ω)=(∫_Ω|∇𝑢|²)^1/2𝑑𝑥, and then 𝐽_𝜆(𝑢) is coercive and bounded from below. It is also easy to see that 𝐽_𝜆(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, min_𝑢∈𝐻¹_0(Ω)𝐽_𝜆(𝑢) is reached at some point 𝑢(𝜆), and 𝑢(𝜆) is a solution of 𝑃_𝜆 when 𝜆∈(0,𝜆₁/𝑎).

Lemma 2. For any 𝜆>𝜆₁/𝑚, 𝑃_𝜆 has no solution, where 𝑚=inf_𝑡>0(𝑓(𝑡)/𝑡).

Proof. If not, assume that 𝑢 is a solution of 𝑃_𝜆 for some 𝜆>𝜆₁/𝑚. Multiplying 𝑃_𝜆 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 𝜆_1Ω𝑢𝜑₁𝑑𝑥=_Ω−Δ𝑢𝜑₁𝑑𝑥=𝜆_Ω𝑓(𝑢)𝜑₁𝑑𝑥>𝜆_1Ω𝑢𝜑₁𝑑𝑥,(6) which is a contradiction.

We begin by show the following.

Lemma 3. There exists a number 𝜆₁/𝑎≤Λ≤𝜆₁/𝑚, such that 𝑃_𝜆 has at least a solution for 𝜆<Λ and has no solution for 𝜆>Λ.

Proof. Let Λ=𝜆∶𝑃_𝜆hasasolution.(7) By Lemmas 1 and 2, 𝜆₁/𝑎≤Λ≤𝜆₁/𝑚. We need just to prove that if (𝑃_𝜇) has a solution, then 𝑃_𝜆 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 (𝑃_𝜇) is a super solution of 𝑃_𝜆 and 𝑢≡0 a subsolution.
It is easy to see that 𝑢_∗≡0 is a subsolution of 𝑃_𝜆, then a standard sub-super solution method’s argument and comparison theorems give the following lemma.

Lemma 4. If 𝑃_𝜆 is solvable, then one has a minimal solution 𝑢_𝜆, that is, for any solution 𝑣 of 𝑃_𝜆, 𝑢_𝜆≤𝑣. Moreover, 𝑢_𝜆 is increasing with respect to 𝜆.

Lemma 5. If 𝜆∈(0,𝜆₁/𝑎), then the solution of  𝑃_𝜆 is unique.

Proof. Suppose that 𝑣₁ and 𝑣₂ are solutions of 𝑃_𝜆. Let 𝑣=𝑣₁−𝑣₂, then −Δ𝑣=𝜆𝑓𝑣₁−𝑓𝑣₂,inΩ,𝑣=0,on𝜕Ω.(8) By mean value theorem, 𝑣 satisfies −Δ𝑣=𝑓^′(𝑣)𝑣,(9) where 𝑣 lies between 𝑣₁ and 𝑣₂. Multiplying 𝑣 and integrating, we get _Ω|∇𝑣|²𝑑𝑥=𝜆_Ω𝑓^′(𝑣)𝑣²𝑑𝑥≤𝑎𝜆_Ω𝑣²𝑑𝑥≤𝑎𝜆𝜆_1Ω|∇𝑣|²𝑑𝑥,(10) which implies that 𝑣≡0. The proof is complete.

Lemma 6. The minimal solution 𝑢_𝜆 is stable, that is, 𝜆₁(−Δ−𝜆𝑓^′(𝑢_𝜆))≥0, where 𝜆₁(−Δ−𝜆𝑓^′(𝑢_𝜆)) denotes the first eigenvalue of the following problem: −Δ𝑤−𝜆𝑓^′𝑢_𝜆𝑤=𝜇𝑤,𝑖𝑛Ω,𝑤=0,𝑜𝑛𝜕Ω.(11)

Proof. Suppose on the contrary that 𝜆₁(−Δ−𝜆𝑓^′(𝑢_𝜆))=𝜇<0, and 𝑤>0 is the corresponding eigenvector. Let 𝑣_𝜀=𝑢_𝜆−𝜀𝜑, then by 𝑃_𝜆 and (11), we have −Δ𝑣_𝜀−𝜆𝑓𝑣_𝜆=𝜆𝑓𝑢_𝜆−𝜆𝜀𝑓^′𝑢_𝜆𝜑−𝜆𝑓𝑢_𝜆−𝜀𝜑−𝜇𝜀𝜑=−𝜇𝜀𝜑+𝑜𝜀𝜑>0,(12) when 𝜀 is small enough, and hence 𝑣_𝜀=𝑢_𝜆−𝜀𝜑 is a super solution of problem 𝑃_𝜆. On the other hand, 0 is a subsolution of 𝑃_𝜆, and Hopf’s boundary lemma implies that 0<𝑣_𝜀 for 𝜀>0 small. An application of sub-sup solution method guarantees that there is a solution 𝑢 of 𝑃_𝜆 satisfying 0<𝑢≤𝑢_𝜆−𝜀𝜑 in Ω, which is a contradiction with the minimality of 𝑢_𝜆. The proof is complete.

Now we state our main result.

Theorem 7. Suppose that 𝑓 satisfies (F1) and (F2), then there exists Λ∈[𝜆₁/𝑎,𝜆₁/𝑚] (where 𝑚=inf_𝑡>0(𝑓(𝑡)/𝑡)) such that problem 𝑃_𝜆(i) has at least one solution for 𝜆∈(0,Λ) and a unique solution for 𝜆∈(0,𝜆₁/𝑎);(ii) has no solution for 𝜆∈(Λ,+∞);(iii)(a) if Λ=𝜆₁/𝑎, then problem 𝑃_𝜆 has no solution at 𝜆=Λ, and lim_{𝜆→Λ−0}𝑢_𝜆(𝑥)=+∞ for all 𝑥∈Ω, where 𝑢_𝜆 denotes the unique solution of 𝑃_𝜆 for 𝜆∈(0,Λ) (see Figure 1), (b) if Λ>𝜆₁/𝑎, then problem 𝑃_𝜆 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 for Λ=𝜆1/𝑎.

Figure 2 

Minimal diagram for Λ>𝜆1/𝑎.

Proof. Statement (i) follows from Lemmas 3 and 5. Statement (ii) follows from Lemma 3. Now we give the proof of statement (iii).
(a) Suppose Λ=𝜆₁/𝑎. The solution 𝑃_𝜆 bifurcates at infinity near Λ=𝜆₁/𝑎 (see [2, 10] for details). On the other hand, 𝑃_𝜆 has a unique solution 𝑢_𝜆 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 lim_{𝜆→Λ−0}𝑢_𝜆(𝑥)=+∞ for all 𝑥∈Ω by the expression of the bifurcation solution in Theorem 13 in Section 3.
If (𝑃_Λ) has a solution, let 𝑢_Λ denote the minimal solution of 𝑃_𝜆. By Lemma 4, 𝑢_𝜆≤𝑢_Λ for 𝜆∈(0,Λ), contradicting lim_{𝜆→Λ−0}‖𝑢_𝜆‖_∞=∞.
(b) For clarity, the proof will be divided into 3 steps.
Step 1. The existence and uniqueness of solutions of 𝑃_𝜆 for 𝜆=𝜆₁/𝑎.
The existence follows directly from Lemma 4. 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 𝑃_𝜆 for 𝜆=Λ.
By Lemmas 3 and 4, 𝑃_𝜆 has a minimal solution 𝑢_𝜆 for any 𝜆∈(0,Λ), and 𝑢_𝜆 is increasing in 𝜆. Let (𝜆_𝑛)⊂(𝜆₁/𝑎,Λ) be any sequence such that lim_𝑛→∞𝜆_𝑛=Λ. Firstly we insure that case (𝑢_𝜆𝑛) is 𝐿²(Ω) bounded. Suppose the contrary that lim_𝑛→∞‖𝑢_𝜆𝑛‖_𝐿²_(Ω)=∞. Let 𝑐_𝑛=‖𝑢_𝜆𝑛‖_𝐿²_(Ω) and 𝑣_𝜆𝑛=𝑢_𝜆𝑛/𝑐_𝑛, then −Δ𝑣_𝜆𝑛=𝜆_𝑛𝑐_𝑛𝑓𝑐_𝑛𝑣_𝜆𝑛𝑣,inΩ,_𝜆𝑛=0,on𝜕Ω.(13)
Since 𝑓(𝑐_𝑛𝑣_𝜆𝑛)/𝑐_𝑛 is bounded in 𝐿²(Ω), it follows from (13) that 𝑣_𝜆𝑛 is bounded in 𝐻¹₀(Ω). Then subject to a subsequence, we may suppose that there exits 𝑣^∗, such that 𝑣_𝜆𝑛⇀𝑣^∗weaklyin𝐻¹₀(Ω),𝑣_𝜆𝑛⟶𝑣^∗stronglyin𝐿²(Ω),𝑣_𝜆𝑛⟶𝑣^∗a.e.inΩ.(14) Then by letting 𝑛→∞, we get from (13) in the weak sense that −Δ𝑣^∗=𝑎Λ𝑣^∗𝑣,inΩ,_∗=0,on𝜕Ω,(15) 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 (𝑢_𝜆𝑛) in 𝐿²(Ω) implies that (𝑢_𝜆𝑛) is bounded in 𝐻¹₀(Ω). Then subject to a subsequence, we may suppose that there exits 𝑢^∗, such that 𝑢_𝜆𝑛⇀𝑢^∗weaklyin𝐻¹₀(Ω),𝑢_𝜆𝑛⟶𝑢^∗stronglyin𝐿²(Ω),𝑢_𝜆𝑛⟶𝑢^∗a.e.inΩ.(16) Then by letting 𝑛→∞, we get −Δ𝑢^∗=Λ𝑓𝑢^∗𝑢,inΩ,_∗=0,on𝜕Ω,(17) and the existence is proved.
Now we prove the uniqueness. Let 𝑢_Λ be the minimal solution of (𝑃_Λ) and 𝑢 a different solution. Then 𝑤∶=𝑢−𝑢_Λ>0 satisfies −Δ𝑣=Λ𝑓^′𝑢_Λ+𝜃𝑤𝑤,inΩ,𝑣=0,on𝜕Ω,(18) where 𝜃∶Ω→ℝ satisfying 0<𝜃<1. It follows that 𝜆₁(−Δ−Λ𝑓^′(𝑢_Λ+𝜃𝑤))=0, where 𝜆₁(−Δ−Λ𝑓^′(𝑢_Λ+𝜃𝑤)) denotes the first eigenvalue of the operator −Δ−Λ𝑓^′(𝑢_Λ+𝜃𝑤) subject to the Dirichlet boundary condition, as defined in Lemma 1. Since 𝑓^′(𝑢_Λ)<𝑓^′(𝑢_Λ+𝜃𝑤) in Ω, we have that 𝜆₁(−Δ−Λ𝑓^′(𝑢_Λ))>𝜆₁(−Δ−Λ𝑓^′(𝑢_𝜆+𝜃𝑤))=0, which implies that the operator −Δ−Λ𝑓^′(𝑢_Λ) is nondegenerate. Then by the Implicit Function Theorem, the solution of 𝑃_𝜆 forms a cure in a neighborhood of (Λ,𝑢_Λ), which is clearly contradicted to the definition of Λ in (7).
Step 3. Prove that 𝑃_𝜆 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 𝑢_𝜆 of 𝑃_𝜆 when 𝜆∈(𝜆₁/𝑎,Λ). Now we must look for another solution 𝑢(>𝑢_𝜆). Assuming that 𝑢=𝑣+𝑢_𝜆, with 𝑣>0, then 𝑣 satisfies −Δ𝑣=𝜆𝑓𝑣+𝑢_𝜆−𝑓𝑢_𝜆,inΩ,𝑣=0,on𝜕Ω.(19) For convenience, let 𝑔(𝑣)=𝑓(𝑣+𝑢_𝜆)−𝑓(𝑢_𝜆) and 𝐺(𝑣)=∫^𝑣₀𝑔(𝑡)𝑑𝑡, then we have −Δ𝑣=𝜆𝑔(𝑣),inΩ,𝑣=0,on𝜕Ω.(20) Define 𝐽_𝜆(𝑣)=_Ω|∇𝑣|²2(𝑣)𝐼−𝜆𝐺𝑑𝑥,_𝜆(𝑣)=_Ω|∇𝑣|²−𝜆𝑣𝑔(𝑣)𝑑𝑥,(21) and the solution manifold 𝑀_𝜆=𝑣∈𝐻¹₀(Ω)∶𝑣>0inΩ,𝐼_𝜆(𝑣)=0.(22)
Firstly we show that 𝑀_𝜆≠𝜙 for any 𝜆∈(𝜆₁/𝑎,Λ). Let 𝜑₁ be the first eigenfunction of −Δ in Ω subject to Dirichlet boundary condition and ∫_Ω𝜑²₁𝑑𝑥=1, then 𝐼_𝜆𝑡𝜑₁=𝜆₁𝑡²−𝜆_Ω𝑡𝜑₁𝑔𝑡𝜑₁𝑑𝑥=𝑡²𝜆₁−𝜆_Ω𝜑₁𝑔𝑡𝜑₁𝑡𝑑𝑥,lim_𝑡→∞Ω𝜑₁𝑔𝑡𝜑₁𝑡𝑑𝑥=lim_𝑡→∞Ω𝜑²₁⋅𝑔𝑡𝜑₁𝑡𝜑₁𝑑𝑥=𝑎.(23)
It follows from (23) that 𝐼_𝜆𝑡𝜑₁<0,(24) for sufficiently large 𝑡 if 𝜆∈(𝜆₁/𝑎,Λ).
On the other hand, let 𝜔₁ be the eigenfunction with ∫_Ω𝜔²₁𝑑𝑥=1 of the first eigenvalue 𝜇₁ of the following equation: −Δ𝜔₁−𝜆𝑓^′𝑢_𝜆𝜔₁=𝜇₁𝜔₁𝜔,inΩ,₁=0,on𝜕Ω.(25)
Since 𝑢_𝜆 is the minimal solution, it follows from Lemmas 4 and 6 that 𝜇₁>0. Then 𝐼_𝜆𝑠𝜔₁=𝑠²_Ω∇𝜔₁²𝑑𝑥−𝜆𝑠_Ω𝜔₁𝑔𝑠𝜔₁𝑑𝑥=𝑠²_Ω∇𝜔₁²𝑑𝑥−𝜆𝑠_Ω𝑓^′𝑢_𝜆𝑠𝜔²₁+𝑜𝑠²𝑑𝑥=𝑠²_Ω∇𝜔₁²−𝜆𝑓^′𝑢_𝜆𝜔²₁𝑑𝑥+𝑜(1)=𝑠²𝜇₁+𝑜(1).(26) Hence 𝐼_𝜆(𝑠𝜔₁)>0 when 𝑠 is small enough. Now it is easy to see that 𝑀_𝜆 is not empty. In fact, take 𝑤_∗=𝑡𝜑₁ for some large 𝑡, and 𝑤^∗=𝑠𝜔 for some small 𝑠>0, such that 𝐼_𝜆𝑤_∗<0,𝐼_𝜆𝑤^∗>0,(27) respectively. Define a continuous function 𝐺 on [0,1], namely, 𝐺(𝜉)=𝐼_𝜆𝜉𝑤_∗+(1−𝜉)𝑤^∗.(28) Then 𝐺(0)>0, 𝐺(1)<0, and hence there exist 𝜉₀∈(0,1) such that 𝐺(𝜉₀)=0, that is, 𝐼_𝜆(𝜉₀𝑤_∗+(1−𝜉₀)𝑤^∗)=0, and 𝑀_𝜆≠𝜙, a desired conclusion.
Since 𝑓 is convex, 𝑔(𝑣) is convex with respect to 𝑣>0 such that 𝑔(𝑣)=𝑔(𝑣)−𝑔(0)≤𝑔^′(𝑣)𝑣.(29)
Integrating (29) with respect to 𝑣 from 0 to 𝑣, we get 2𝐺(𝑣)≤𝑔(𝑣)𝑣.(30) Therefore, on 𝑀_𝜆𝐽_𝜆(𝑣)=𝜆2_Ω𝑔(𝑣)𝑣−2𝐺(𝑣)𝑑𝑥≥0,(31) that is, 𝐽_𝜆(𝑣) is bounded from below.
And then we obtain a nonminimal positive solution of 𝑃_𝜆 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 Theorem 7, 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 lim_𝑡→∞𝐿(𝑡)=𝐿_∞ exists (may be +∞). Also note that 𝐿(0)=−𝑓(0)<0.

Theorem 9. If 𝐿_∞≤0, then Λ=𝜆₁/𝑎; if 𝐿_∞>0, then Λ>𝜆₁/𝑎.

Proof. (i) If 𝐿_∞≤0, then 𝑓(𝑡)≥𝑎𝑡 for all 𝑡≥0. We prove that 𝑃_𝜆 has no solution and hence Λ=𝜆₁/𝑎. Suppose the contrary that 𝑢 is a solution 𝑃_𝜆 for 𝜆=𝜆₁/𝑎, then 𝜆−Δ𝑢=¹𝑎𝑓(𝑢)≥𝜆₁𝑢.(32) Let 𝜑 be a positive eigenfunction of the first eigenvalue 𝜆 of −Δ on Ω with Dirichlet boundary condition, that is Δ𝜑+𝜆₁𝜑=0,inΩ,𝜑=0,on𝜕Ω.(33) Multiplying (32) by 𝜑>0, and integrating by parts, we get _Ω𝑓(𝑢)−𝑎𝑢𝜑𝑑𝑥=0,(34) which yields that 𝑓(𝑢)=𝑎𝑢, contradicting the fact that 𝑓(0)>0.
(ii) If 𝐿_∞>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 (𝑠)(𝑠)𝑓(𝑢(𝑠𝑢(𝑠)Δ𝑢+𝜆))=0,inΩ,=0,on𝜕Ω.(35)
It follows from (33) and (35) that 𝜆(𝑠)_Ω𝑓(𝑢(𝑠))𝜑𝑑𝑥=𝜆_1Ω𝑢(𝑠)=𝜆𝜑𝑑𝑥₁𝑎_Ω𝑎𝑢(𝑠)𝜑𝑑𝑥.(36) By the fact that 𝑢(𝑠)(𝑥)=𝑠𝜑(𝑥)+𝑧(𝑠)(𝑥)→∞ (𝑠→∞) a.e. in Ω, we have 𝑎𝑢(_Ω𝑠)𝜑𝑑𝑥−_Ω𝑓(𝑢(𝑠=))𝜑𝑑𝑥_Ω𝑎𝑢(𝑠)−𝑓(𝑢(𝑠))𝜑𝑑𝑥>0,(37) 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 𝐾(𝑡)=𝑡𝑓^′(𝑡)−𝑓(𝑡),(38) then 𝐾^′(𝑡)=𝑡𝑓^′′(𝑡)>0 a.e. in (0,+∞), and 𝐾(𝑡) is strictly increasing, and 𝐾(0)=−𝑓(0)<0. Denote lim_𝑡→∞𝐾(𝑡)=𝐾_∞∈(−∞,+∞].(39)

Theorem 10. If 𝐾_∞≤0, then Λ=𝜆₁/𝑎; if 𝐾_∞>0, then Λ>𝜆₁/𝑎.

Proof. If 𝐾_∞≤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, if 𝐾_∞>0, by 𝐿(𝑡)−𝐾(𝑡)=𝑡𝑎−𝑓^′(𝑡)>0,∀𝑡>0,(40) 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 Theorem 7, the exact number of solutions 𝑃_𝜆 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 𝑃_𝜆 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 that Ω is the unit ball in 𝑅^𝑁 centered with the origin.

The next remarkable results regarding 𝑃_𝜆 are due to Gidas et al. [15] and Lin and Ni [16].

Lemma 11. (1) If 𝑓 is locally Lipschitz continuous in [0,∞), then all positive solutions of 𝑃_𝜆 are radially symmetric, that is, 𝑢(𝑥)=𝑢(𝑟), 𝑟=|𝑥|, and satisfies 𝑢^′′+𝑛−1𝑟𝑢^′(𝑢)(),𝑢+𝜆𝑓=0,𝑟∈0,1_′(0)=𝑢(1)=0.(41) Moreover, 𝑢^′(𝑟)<0 for all 𝑟∈(0,1], and hence 𝑢(0)=max_0≤𝑟≤1𝑢(𝑟).
(2) Suppose 𝑓∈𝐶¹(𝑅). If 𝑢 is a positive solution to 𝑃_𝜆, and 𝑤 is a solution of the linearized problem (43) (if it exists), then 𝑤 is also radially symmetric and satisfies 𝑤^′′+𝑛−1𝑟+𝜆𝑓^′(𝑢)(),𝑤𝑤=0,𝑟∈0,1_′(0)=𝑤(1)=0.(42)

The next lemma also plays a key role in this section.

Lemma 12. (1) For any 𝑑>0, there is at most one 𝜆_𝑑>0 such that 𝑃_𝜆 have a positive solution 𝑢(⋅) with 𝜆=𝜆_𝑑 and 𝑢(0)=𝑑.
(2) Let 𝑇={𝑑>0∶(𝑃_𝜆) have a positive solution with  𝑢(0)=𝑑}, then 𝑇 is open; 𝜆(𝑑)=𝜆_𝑑 is a well-defined continuous function from 𝑇 to 𝑅⁺.

Lemma 12 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]). Suppose 𝑓∈𝐶¹(𝑅). Let 𝑙𝑖𝑚_𝑢→∞𝑓(𝑢)/𝑢=𝑎∈(0,∞) and 𝜆_∞=𝜆₁/𝑎. Then all positive solutions of 𝑃_𝜆 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, 𝑙𝑖𝑚_𝑠→∞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.

Let 𝑢 be a solution of problem 𝑃_𝜆, then 𝑢 is called a degenerate solution if the corresponding linearized equation −Δ𝑤=𝜆𝑓^′(𝑢)𝑤,inΩ,𝑤=0,on𝜕Ω,(43) has a nontrivial solution.

Now suppose that 𝑓 satisfies (F1), (F2). As in the end of Section 2, let 𝐾(𝑡)=𝑡𝑓^′(𝑡)(𝑡)𝐾−𝑓_∞=lim_𝑡→∞𝐾(𝑡).(44) If 𝐾_∞>0, then there exists a unique real number 𝛽>0, such that 𝐾(𝑡);𝐾(𝑡)𝛽<0for𝑡∈0,𝛽>0for𝑡∈𝛽,∞;𝐾=0.(45)

Lemma 14. Suppose that 𝐾_∞>0. If 𝑢 is a degenerate solution of 𝑃_𝜆, then 𝑢(0)>𝛽.

Proof . Suppose the contrary that 𝑢(0)≤𝛽, then 𝐾(𝑢)=𝑢𝑓^′(𝑢)−𝑓(𝑢)<0,inΩ⧵{0}.(46) Let 𝑤 be a nontrivial solution of the corresponding linearized equation (43). From 𝑃_𝜆 and (43), we get 0=_Ω(−Δ𝑤𝑢+Δ𝑢𝑤)𝑑𝑥=𝜆_Ω𝑢𝑓^′(𝑢)−𝑓(𝑢)𝑤𝑑𝑥.(47) It appears from (46) and (47) that 𝑤 must change sign in Ω.
In view of Lemma 11(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 _{Ω⧵𝐵(𝑟1)}(−Δ𝑤𝑢+Δ𝑢𝑤)𝑑𝑥=𝜆_Ω𝑢𝑓^′(𝑢)−𝑓(𝑢)𝑤𝑑𝑥<0,(48) where 𝐵(𝑟₁) denotes the ball of radius 𝑟₁ centered with the origin.
On the other hand, using integration by parts, we have _{Ω⧵𝐵(𝑟1)}(−Δ𝑤𝑢+Δ𝑢𝑤)𝑑𝑥=−_{𝜕(Ω⧵𝐵(𝑟1))}𝜕𝑤𝜕𝜈𝑢𝑑𝑠>0.(49) a contradiction.

Theorem 15. Suppose that 𝑓 satisfies (F1)-(F2) with 0<𝐾_∞<𝑎𝛽. If 𝑢 is a degenerate solution of 𝑃_𝜆, then any nontrivial solution of the corresponding linearized equation (43) does not change sign in Ω.

Proof . By Lemma 14, max_𝑥∈Ω𝑢(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 that 𝑤(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 that 𝑤(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 𝑃_𝜆 by 𝑢−𝛽, (43) by 𝑤, subtracting, and integrating on 𝐵(𝑟₀), we get _𝐵(𝑟0)−Δ𝑤𝑢−𝛽+Δ𝑢𝑤𝑑𝑥=𝜆_𝐵(𝑟0)𝑢−𝛽𝑓^′(𝑢)−𝑓(𝑢)𝑤𝑑𝑥.(50)
Let 𝐽(𝑡)=(𝑡−𝛽)𝑓^′(𝑡)−𝑓(𝑡), then 𝐽(0)=−𝑓(0)<0, 𝐽(∞)=lim_𝑡→∞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 𝑢−𝛽𝑓_𝐵(𝑟0)^′(𝑢)−𝑓(𝑢)𝑤𝑑𝑥<0.(51)
On the other hand, by Green formula, _𝐵(𝑟0)−Δ𝑤𝑢−𝛽+Δ𝑢𝑤𝑑𝑥=−_{𝜕(𝐵(𝑟0))}𝜕𝑤𝜕𝜈𝑢−𝛽𝑑𝑥>0.(52) 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 define 𝐹∶𝐶₀^2,𝛼(Ω)→𝐶^𝛼(Ω), by 𝐹𝑢=Δ𝑢+𝜆𝑓(𝑢),(53) then the linearized operator (Frechèt derivative) is 𝐹_𝑢(𝜆,𝑢)𝑤=Δ𝑤+𝜆𝑓^′(𝑢)𝑤.(54)

From the maximum principle, all solutions of 𝑃_𝜆 are positive on Ω. Moreover, if (𝜆^∗,𝑢^∗) is degenerate solution of 𝑃_𝜆, 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(𝐹_𝑢(𝜆^∗,𝑢^∗))=span{𝑤}, and it follows from Fredholm alternative theorem that codim𝑅(𝐹_𝑢(𝜆^∗,𝑢^∗))=1. Now we prove that 𝐹_𝜆(𝜆^∗,𝑢^∗)∉𝑅(𝐹_𝑢(𝜆^∗,𝑢^∗)). If it is not the case, then there exists 𝑣∈𝐶₀^2,𝛼(Ω), such that Δ𝑣+𝜆^∗𝑓^′𝑢^∗𝑣=𝑓𝑢^∗.(55)

We also have Δ𝑤+𝜆^∗𝑓^′𝑢^∗𝑤=0.(56)

Multiplying (55) by 𝑤, (56) by 𝑣, subtracting, and integrating, we obtain _Ω𝑓𝑢^∗𝑤𝑑𝑥=0,(57) a contradiction. As all the conditions of Crandall-Rabinowitz’s bifurcation theorem [20] are satisfied, the solutions of 𝑃_𝜆 near the degenerate solution (𝜆^∗,𝑢^∗) form a smooth curve which is expressed in the form (𝜆(𝑠),𝑢(𝑠))=𝜆^∗+𝜏(𝑠),𝑢₀+𝑠𝑤+𝑧(𝑠),(58) 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.

Substituting 𝑢 and 𝜆 by expression (58), then differentiating the equation 𝑃_𝜆 twice, and evaluating at 𝑠=0, we have Δ𝑢_𝑠𝑠+𝜆𝑓(𝑢)𝑢_𝑠𝑠+2𝜆^′𝑓^′(𝑢)𝑢_𝑠+𝜆𝑓^′′(𝑢)𝑢²_𝑠+𝜆^′′𝑓(𝑢)=0,Δ𝑢_𝑠𝑠+𝜆^∗𝑓^′(𝑢)𝑢_𝑠𝑠+𝜆^∗𝑓^′′(𝑢)𝑤²+𝜆^′′(0)𝑓(𝑢)=0.(59) Multiplying (59) by 𝑤, (43) by 𝑢_𝑠𝑠, subtracting, and integrating, we obtain 𝜏^′′(0)=−𝜆^∗∫_Ω𝑓^′′𝑢^∗𝑤³𝑑𝑥∫_Ω𝑓(𝑢^∗)𝑤𝑑𝑥<0.(60)

By (60) and the Taylor expansion formula of 𝜏(𝑠) at 𝑠=0, we conclude that at any degenerate solution (𝜆^∗,𝑢^∗) of 𝑃_𝜆, 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 that Ω is the unit ball in 𝑅^𝑛, 𝑓 satisfies (F1)-(F2), and 0<𝐾_∞<𝑎𝛽. Then for problem 𝑃_𝜆, (1)there exist no solutions for 𝜆>Λ, (2)there exists exactly one solution for 𝜆∈(0,𝜆₁/𝑎]∪{Λ}, (3)there exist exactly two solutions for 𝜆∈(𝜆₁/𝑎,Λ).
Moreover, the solution set {(𝜆,𝑢)} of 𝑃_𝜆 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 Theorem 10, Λ>𝜆₁/𝑎, and Theorem 7 tells us that 𝑃_𝜆 has a unique solution (Λ,𝑢_Λ) for 𝜆=Λ, and Implicit Function Theorem implies that (Λ,𝑢_Λ) 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,𝑢_Λ) 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)>𝑢_Λ(0) the upper branch, and the rest the lower branch. We denote the upper branch by 𝑢^𝜆 and the lower branch by 𝑢_𝜆.
For the upper branch, as long as (𝜆,𝑢^𝜆) 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,𝑢^𝜆). This process of continuation towards smaller values of 𝜆 will not encounter any other degenerate solutions. This is because, if, say, (𝜆,𝑢^𝜆) becomes degenerate at 𝜆=𝜆₀, the discussion prior to this theorem implies that all the solutions near (𝜆₀,𝑢^𝜆0) must lie to the left side of it, which is a contradiction. Lemma 12 tells us that 𝜆→𝑢^𝜆(0) is decreasing. So in the progress of extension of (𝜆,𝑢^𝜆) towards smaller values of 𝜆, there are only the following two possibilities.(i)The upper branch (𝜆,𝑢^𝜆) stops at some (0,𝑢₀), and 𝑢₀(0)>𝑢_Λ(0). (ii)‖𝑢_𝜆‖_∞ goes to infinity as 𝜆→𝜆^∼+0,  0≤𝜆^∼<Λ.
But case (i) cannot happen, since (0,𝑢₀) is obviously not a solution of  𝑃_𝜆. Hence case (ii) happens. We assert that 𝜆^∼=𝜆₁/𝑎. In fact, let {𝜆_𝑛} be an arbitrary sequence such that 𝜆_𝑛→𝜆^∼. Denote 𝑀_𝑛=‖𝑢_𝑛‖_∞, 𝑣_𝑛=𝑢_𝑛/𝑀_𝑛, then 𝑀_𝑛→∞ and Δ𝑣_𝑛+𝜆_𝑛𝑓𝑀_𝑛𝑣_𝑛𝑀_𝑛=0,inΩ,𝑣=0,on𝜕Ω.(61) Since 𝑓(𝑀_𝑛𝑣_𝑛)/𝑀_𝑛 is bounded, by Sobolev Imbedding Theorems and standard regularity of elliptic equation, it is easy to see that {𝑣_𝑛} has a subsequence, still denoted by {𝑣_𝑛}, such that 𝑣_𝑛→𝑣 in 𝐶^2,𝛼(Ω)(𝑛→∞), for some 𝑣∈𝐶^2,𝛼(Ω), 𝑣>0 in Ω. Letting 𝑛→∞ in (61), we get Δ𝑣+𝜆^∼𝑎𝑣=0,inΩ,𝑣=0,on𝜕Ω,(62) 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,𝑢_𝜆) 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,𝑢_𝜆). This process of continuation towards smaller values of 𝜆 will not encounter any other degenerate solutions. Lemma 12 implies that 𝜆→𝑢_𝜆(0) is increasing. So in the progress of extension of (𝜆,𝑢_𝜆) towards smaller values of 𝜆, there are only the following two possibilities.(i)The lower branch (𝜆,𝑢_𝜆) stops at some (0,𝑢₀) with 𝑢₀(0)>0.(ii)The lower branch (𝜆,𝑢_𝜆) stops at some (𝜆₀,0) with 0≤𝜆₀<Λ.
As before, case (i) will not happen. Then case (ii) happens. By 𝑓(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,𝑢^𝜆)} and a lower branch {(𝜆,𝑢_𝜆)}. The lower branch starts from (Λ,𝑢_Λ) and stops at (0,0), and 𝜆→𝑢_𝜆(0) is a strictly increasing function. The upper branch {(𝜆,𝑢^𝜆)} starts from (Λ,𝑢_Λ) and stops at (𝜆₁/𝑎,∞), and 𝜆→𝑢^𝜆(0) is a strictly decreasing function with 𝑢^𝜆(0) blowing up as 𝜆→𝜆₁/𝑎+0. By Lemma 12, all solutions of 𝑃_𝜆 are contained in this smooth solution curve, and the complete bifurcation diagram can be described as in Figure 3. The proof is complete.

Acknowledgment

The author thanks the reviewer for his/her comments which helped to improve the content of the paper.