Positive Solutions of Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

Abstract

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 ℝ^𝑁, 𝑁≥2, with smooth boundary 𝜕Ω. The nonlinear elliptic boundary value problem is defined as −Δ𝑢=𝜆𝑎(𝑥)𝑓(𝑢)inΩ,𝜕𝑢(𝑥)𝑔(𝑢)𝜕𝐧+𝑏=0on𝜕Ω, (1) stimulated by a problem of chemical reactor theory [6], where Δ=∑^𝑁_𝑗=1(𝜕²/𝜕𝑥²_𝑗), 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(∞)=lim_𝑢→∞(𝑓(𝑢)/𝑢), 𝑔_∞=lim_𝑢→∞(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 liminf_𝑠→∞𝑓(𝑠)<limsup_𝑠→∞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:(A1)𝑎∈𝐶^𝜃(Ω) with 𝑎(𝑥)>0 in 𝑥∈Ω; 𝑏∈𝐶^1+𝜃(𝜕Ω) with 𝑏≥0 and 𝑏≢0 on 𝜕Ω;(A2)𝑓∈𝐶¹([0,∞)) and there exist constants 𝑓_∞,𝑓^∞∈(0,∞) and functions ℎ₁,ℎ₂∈𝐶¹([0,∞)), such that 𝑓_∞=liminf_𝑠→∞𝑓(𝑠)𝑠,𝑓^∞=limsup_𝑠→+∞𝑓(𝑠)𝑠,𝑓_∞𝑠+ℎ₁(𝑠)≤𝑓(𝑠)≤𝑓^∞𝑠+ℎ₂(𝑠),𝑠∈[0,∞), (2) with ℎ_𝑗(𝑠)=𝑜(|𝑠|)as𝑠⟶∞,𝑗=1,2; (3)(A3)𝑔∈𝐶¹([0,∞)) and there exist constants 𝑔_∞∈(0,∞) and functions 𝑘∈𝐶([0,∞)), such that 𝑔(𝑠)=𝑔_∞𝑠+𝑘(𝑠),𝑘(𝑠)=𝑜(|𝑠|)as𝑠⟶+∞. (4)

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

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

Definition 1 (see [12, page 450]). A solution set 𝒮 of (1) is said to bifurcate from infinity in the interval [𝑎,𝑏], if(i)the solutions of (1) are, a priori, bounded in 𝑋 for 𝜆=𝑎 and 𝜆=𝑏,(ii)there exists {(𝜆_𝑛,𝑢_𝑛)}⊂𝒮 such that {𝜆_𝑛}⊂[𝑎,𝑏] and ‖𝑢_𝑛‖→∞.

By a constant 𝜆₁ we denote the first eigenvalue of the eigenvalue problem: −Δ𝜑=𝜆𝑎(𝑥)𝜑inΩ,𝜕𝜑(𝑥)𝑔𝜕𝐧+𝑏_∞𝜑=0on𝜕Ω. (6) 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∈𝐶^2+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 liminf_𝑢→∞ℎ₁(𝑢)>𝑓_∞𝑔_∞limsup_𝑢→∞𝑘(𝑢),𝑟𝑒𝑠𝑝.,limsup_𝑢→∞ℎ₂(𝑢)<𝑓^∞𝑔_∞liminf_𝑢→∞𝑘(𝑢). (7)(8) 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 𝑢=𝐹(𝜆,𝑢). (9)

Lemma 4 (see [12, 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 𝑢≠𝐹(𝑎,𝑢),𝑢≠𝐹(𝑏,𝑢),∀𝑢∶‖𝑢‖≥𝑅. (10) Furthermore, assume that 𝑑𝐼−𝐹(𝑎,⋅),𝐵_𝑅(0),0≠𝑑𝐼−𝐹(𝑏,⋅),𝐵_𝑅(0),0 (11) 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(i)𝒞 is unbounded in 𝜆 direction or else(ii)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∶𝐶^𝜃(Ω)→𝐶^2+𝜃(Ω) be the resolvent of the linear boundary value problem: −Δ𝑢=𝜙inΩ,𝜕𝑢(𝑥)𝑔𝜕𝐧+𝑏_∞𝑢=0on𝜕Ω. (12) 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∶𝐶^1+𝜃(𝜕Ω)→𝐶^2+𝜃(Ω) be the resolvent of the linear boundary value problem: −Δ𝑢=0inΩ,𝜕𝑢(𝑥)𝑔𝜕𝐧+𝑏_∞𝑢=𝜓on𝜕Ω. (13) 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: 𝑢=𝜆𝒦𝑎𝑓(𝑢)+ℛ[𝑏𝜏(−𝑘(𝑢))]in𝐶Ω. (14) 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 (𝜆_𝑗,𝑢_𝑗) be nonnegative solutions of (1) with 𝜆=𝜆_𝑗 such that 𝑢_𝑗⟶∞as𝑗⟶∞,𝜆_𝑗∈𝛼,𝛽. (15) If 𝑣_𝑗=𝑢_𝑗𝑢_𝑗, (16) then we have 𝑣_𝑗=𝜆_𝑗𝒦𝑎𝑓𝑢_𝑗𝑢_𝑗−𝑘𝑢+ℛ𝑏𝜏^𝑗𝑢_𝑗in𝐶Ω. (17) From conditions (3) and (6), for any 𝜖>0, there exist constants 𝑑_𝜖,𝑐_𝜖>0 such that 𝑓(𝑠)−𝑓^∞𝑠≤𝜖𝑠+𝑑_𝜖𝑓(𝑠),∀𝑢≥0,−𝑓_∞𝑠≥−𝜖𝑠−𝑑_𝜖|𝑘(𝑠,∀𝑢≥0,)|≤𝜖𝑠+𝑐_𝜖,∀𝑢≥0. (18)(19)(20) This implies that both 𝑓(𝑢_𝑗)/‖𝑢_𝑗‖ and 𝑘(𝑢_𝑗)/‖𝑢_𝑗‖ 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 {𝑣_𝑗}, still denoted by {𝑣_𝑗}, such that 𝑣_𝑗⟶𝑣₀in𝐶Ωas𝑗⟶∞. (21) By (15) it follows from (18)–(20) that limsup_𝑗→∞max_𝑥∈Ω𝑓𝑢_𝑗𝑢_𝑗−𝑓^∞𝑣_𝑗≤𝜖,liminf_𝑗→∞min_𝑥∈Ω𝑓𝑢_𝑗𝑢_𝑗−𝑓_∞𝑣_𝑗≥−𝜖,limsup_𝑗→∞𝑘𝑢_𝑗𝑢_𝑗≤𝜖. (22) Since 𝜖 is arbitrary, it follows that limsup_𝑗→∞𝑓𝑢_𝑗𝑢_𝑗≤𝑓^∞𝑣₀in𝐶Ω,liminf_𝑗→∞𝑓𝑢_𝑗𝑢_𝑗≥𝑓_∞𝑣₀in𝐶Ω. (23) Let 𝜆_𝑗→𝜆 and 𝑓(𝑢_𝑗)/‖𝑢_𝑗‖→𝑤₀ as 𝑗→∞. Then in view of (17), 𝑣₀=𝜆𝒦𝑎𝑤₀. (24)
We claim that 𝜆𝜆∈¹𝑓^∞,𝜆₁𝑓_∞. (25)
Since 𝑓_∞𝑣₀≤𝑤₀≤𝑓^∞𝑣₀, (26) it follows from (26) that 𝜆𝒦𝑎𝑓_∞𝑣₀≤𝑣₀≤𝜆𝒦𝑎𝑓^∞𝑣₀, (27) which implies 𝜆𝑓_∞𝜆₁𝜆𝑓≤1≤^∞𝜆₁. (28) That is, 𝜆₁𝑓^∞≤𝜆𝜆≤¹𝑓_∞. (29)
Since ‖𝑣₀‖=1 and 𝑣₀≥0, the strong positivity of 𝒦 ensures that 𝑣₀>0 on Ω, and accordingly, 𝑣_𝑗>0 on Ω for 𝑗 large enough and so is 𝑢_𝑗 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): Φ(𝜆,𝑢)∶=𝑢−𝜆𝒦𝑎𝑓(|𝑢|)+ℛ[𝑏𝜏(𝑘(|𝑢|))]. (30) 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(i)the solutions of (30) are, a priori, bounded in 𝑋 for 𝜆=𝜆₁/𝑓^∞−𝜎 and 𝜆=𝜆₁/𝑓_∞+𝜎,(ii)there exists {(𝜆_𝑛,𝑢_𝑛)}⊂𝒮 such that {𝜆_𝑛}⊂[𝜆₁/𝑓^∞−𝜎,𝜆₁/𝑓_∞+𝜎] and ‖𝑢_𝑛‖→∞.

Let {𝜆_𝑛𝑗} be any convergent subsequence of {(𝜆_𝑛,𝑢_𝑛)}, and let lim_𝑗→∞𝜆_𝑛𝑗=𝜆^♯. (31)

We claim that 𝜆^♯∈𝜆₁𝑓^∞,𝜆₁𝑓_∞and𝑢_𝑛𝑗>0onΩ,if𝑗islargeenough. (32)

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 𝑣₀=𝜆^♯𝒦𝑎𝑤₀. (33) This together with the strong positivity of 𝒦 implies that 𝑣₀>0,onΩ. (34) Since 𝑓_∞𝑣₀≤𝑤₀≤𝑓^∞𝑣₀, (35) it follows from (34) that 𝜆^♯𝒦𝑎𝑓_∞𝑣₀≤𝑣₀≤𝜆^♯𝒦𝑎𝑓^∞𝑣₀, (36) which implies𝜆^♯𝑓_∞𝜆₁≤1≤𝜆^♯𝑓^∞𝜆₁. (37) That is, 𝜆₁𝑓^∞≤𝜆^♯≤𝜆₁𝑓_∞. (38)

From (34), it follows that 𝑣_𝑗>0 on Ω for 𝑗 large enough and so is 𝑢_𝑗 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.

LetΦ_𝑝∶(0,∞)×𝐶(Ω)→𝐶(Ω)be defined as Φ_𝑝(𝜆,𝑢)∶=𝑢−𝜆𝒦𝑎𝑓(|𝑢|)+𝑝(𝜆)ℛ[𝑏𝜏(𝑘(|𝑢|))]. (39) Here 𝑝∶[0,𝜆₁/𝑓^∞]→[0,1] is a smooth cut-off function such that 𝑝(𝜆)=𝜆0near𝜆=0,1near𝜆=¹𝑓^∞. (40)

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 Φ_𝑝𝜇,𝑢≠0,𝜇∈Λ,𝑢∈𝐶Ω∶‖𝑢‖≥𝑟. (41)

Proof. Assume on the contrary that there exist 𝜇_𝑗≥0, 𝑢_𝑗∈𝐶(Ω), and 𝜇₀∈Λ such that Φ_𝑝𝜇_𝑗,𝑢_𝑗𝜇=0,_𝑗⟶𝜇₀,𝑢_𝑗⟶∞,as𝑗⟶∞. (42) 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 Φ(𝜆,𝑢)≠𝑡𝜑₁,𝑡∈[0,1],𝑢∈𝐶Ω∶‖𝑢‖≥𝑟. (43)

Proof. Assume on the contrary that there exist 𝜆₀∈(𝜆₁/𝑓_∞,∞), 𝑡₀,𝑡_𝑗∈[0,1], and 𝑢_𝑗∈𝐶(Ω) which can be taken such that Φ𝜆₀,𝑢_𝑗=𝑡_𝑗𝜑₁,𝑡_𝑗⟶𝑡₀,𝑢_𝑗⟶∞,as𝑗⟶∞. (44) Using the same argument as that in the proof of Proposition 5, we can obtain a subsequence of {𝑢_𝑗}, still denoted by {𝑢_𝑗}, which may satisfy that 𝑢_𝑗>0 on Ω for all 𝑗≥1. It follows that 𝑢_𝑗=𝜆₀𝒦𝑎𝑓𝑢_𝑗−ℛ𝑏𝜏𝑘𝑢_𝑗+𝑡_𝑗𝜑₁,𝑡_𝑗⟶𝑡₀∈[0,1],𝑢_𝑗⟶∞as𝑗⟶∞. (45)
For a function 𝜑 we let ⟨𝜑⟩={𝑠𝜑∶𝑠∈ℝ}. The projection theorem derives the orthogonal decomposition of the Lebesgue space 𝐿²(Ω) as 𝐿²(Ω)=𝜑₁⊕𝑊;𝑢=𝑠𝜑₁+𝑤. (46) 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 𝑢_𝑗 is given as 𝑢_𝑗=𝑠_𝑗𝜑₁+𝑤_𝑗,𝑠_𝑗=_Ω𝑢_𝑗𝜑₁𝑑𝑥>0,𝑤_𝑗∈𝑊. (47) By the regularity argument, (45) gives that 𝑢_𝑗∈𝐶^2+𝜃(Ω), and thus −Δ𝑢_𝑗=𝜆₀𝑎𝑓𝑢_𝑗+𝑡_𝑗𝜆₁𝑎𝜑₁inΩ,𝜕𝑢_𝑗(𝑥)𝑔𝜕𝐧+𝑏_∞𝑢_𝑗=−𝑏(𝑥)𝑘𝑢_𝑗on𝜕Ω. (48) By Green's formula it follows that _Ω𝜆₀𝑎𝑓𝑢_𝑗+𝑡_𝑗𝜆₁𝑎𝜑₁𝜑₁−𝜆₁𝑎𝑢_𝑗𝜑₁=𝑑𝑥_Ω−Δ𝑢_𝑗𝜑₁+𝑢_𝑗Δ𝜑₁𝑑𝑥=_𝜕Ω𝑏𝑘𝑢_𝑗𝜑₁𝑑𝜎. (49) Here 𝑑𝜎 is the surface element of 𝜕Ω. This implies that 𝑏𝑘𝑢_𝜕Ω𝑗𝜑₁𝑑𝜎≥𝜆₀𝑓_∞−𝜆_1Ω𝑎𝑢_𝑗𝜑₁𝑑𝑥+𝜆_0Ω𝑎ℎ₁𝑢_𝑗𝜑₁𝑑𝑥. (50) Hence assertion (19) gives _𝜕Ω𝑏𝑘𝑢_𝑗𝑢_𝑗𝜑₁𝑑𝜎≥𝑓_∞𝜆₀−𝜆₁+𝜆₀𝜖𝑓_∞×_Ω𝑎𝑢_𝑗𝜑₁𝑑𝑥𝑢_𝑗−𝜆₀‖𝑎‖𝑑_𝜖𝜑_1𝐿1(Ω)𝑢_𝑗. (51) Now use again for (48) the same procedure as in the proof of Proposition 5; then we see that some subsequence of {𝑢_𝑗/‖𝑢_𝑗‖}, still denoted by {𝑢_𝑗/‖𝑢_𝑗‖}, 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 0=lim_{𝑗→∞𝜕Ω}𝑏𝑘𝑢_𝑗𝑢_𝑗𝜑₁𝑑𝜎≥lim_𝑗→∞𝑠_𝑗𝑢_𝑗𝑓_∞𝜆₀−𝜆₁+𝜆₀𝜖𝑓_∞−𝜆₀‖𝑎‖𝑑_𝜖𝜑_1𝐿1(Ω)𝑢_𝑗=𝑓_∞𝜆₀−𝜆₁+𝜆₀𝜖𝑓_∞Ω𝑎𝑣₀𝜑₁𝑑𝑥>0. (52)

Lemma 8. Let 𝜆⁻_𝑛=𝜆₁/𝑓^∞−1/𝑛, and let 𝜆⁺_𝑛=𝜆₁/𝑓_∞+1/𝑛, where 𝑛∈ℕ satisfies 𝑛≥𝑓^∞/𝜆₁. Assume that (A1)–(A3) hold. Then there exists constant 𝑟_𝑛>0 satisfying that 𝑟_𝑛→∞ as 𝑛→∞, such that for any 𝑛 large enough degΦ𝜆⁻_𝑛,⋅,𝐵_𝑟𝑛Φ𝜆,0=1,deg⁺_𝑛,⋅,𝐵_𝑟𝑛,0=0. (53)(54)

Proof. First we show assertion (53). From Lemma 6, there exists 𝑟_𝑛>0 such that 𝑟_𝑛→∞ as 𝑛→∞ satisfying that Φ_𝑝(𝜆,𝑢)≠0,∀𝜆∈0,𝜆⁻_𝑛,∀𝑢∈𝐶Ω∶‖𝑢‖=𝑟_𝑛. (55) Since 𝑝(0)=0 and 𝑝(𝜆⁻_𝑛)=1 for 𝑛 large enough from (40), by the homotopy invariance and normalization it follows that for any 𝑛 large enough degΦ𝜆⁻_𝑛,⋅,𝐵_𝑟𝑛,0=degΦ_𝑝𝜆⁻_𝑛,⋅,𝐵_𝑟𝑛Φ,0=deg_𝑝(0,⋅),𝐵_𝑟𝑛𝐼,0=deg_𝐶(Ω),𝐵_𝑟𝑛,0=1. (56)
Next, we show assertion (54). We may derive from Lemma 7 that Φ𝜆⁺_𝑛,𝑢≠𝑡𝜑₁,∀𝑡∈[0,1],∀𝑢∈𝐶Ω∶‖𝑢‖≥𝑟_𝑛. (57) So for any 𝑛 large enough, by the homotopy invariance, it follows that degΦ𝜆⁺_𝑛,⋅,𝐵_𝑟𝑛,0=degΦ𝜆⁺_𝑛,⋅−𝜑₁,𝐵_𝑟𝑛,0=0. (58)

Proof of Theorem 2. For any fixed 𝑛∈ℕ with 𝜆₁/𝑓^∞−1/𝑛>0, set 𝛼_𝑛=𝜆₁/𝑓^∞−1/𝑛 and 𝛽_𝑛=𝜆₁/𝑓_∞+1/𝑛. It is easy to verify that, for any fixed 𝑛 large enough, there exists 𝑟_𝑛 such that 𝑟_𝑛→∞ as 𝑛→∞ satisfying that, for any 𝑟≥𝑟_𝑛, it follows from Lemmas 6–8 that all conditions of Lemma 4 are satisfied. So there exists a closed connected component 𝒞_𝑛 of solutions (14) that is unbounded in [𝛼_𝑛,𝛽_𝑛]×𝐶(Ω) and either(i)𝒞_𝑛 is unbounded in 𝜆 direction or else(ii)there exists an interval [𝑐,𝑑] such that (𝛼_𝑛,𝛽_𝑛)∩(𝑐,𝑑)=∅ and 𝒞_𝑛 bifurcates from infinity in [𝑐,𝑑]×𝐶(Ω).
By Lemma 6, the case (ii) cannot occur. Thus 𝒞_𝑛 is unbounded bifurcated from [𝛼_𝑛,𝛽_𝑛]×∞ in ℝ×𝐶(Ω). Furthermore, we have from Lemma 6 that, for any closed interval 𝐼⊂[𝛼_𝑛,𝛽_𝑛]⧵[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 𝒞_𝑛 must be bifurcated from [𝜆₁/𝑓^∞,𝜆₁/𝑓_∞]×{∞}.

Proof of Theorem 3. Under condition (7), assume to the contrary that there exist positive solutions 𝑢_𝑗 of (1) with 𝜆=𝜆_𝑗≥𝜆₁/𝑓_∞ such that 𝜆_𝑗→𝜆₁/𝑓_∞ and ‖𝑢_𝑗‖→∞ as 𝑗→∞. If 𝑣_𝑗=𝑢_𝑗/‖𝑢_𝑗‖, 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 {𝑣_𝑗}, still denoted by {𝑣_𝑗}, tends to 𝑣₀ in 𝐶(Ω). It follows that for any 𝑗 large enough we have 𝑣_𝑗(𝑥)>min_𝑥∈Ω𝑣₀(𝑥)2onΩ, (59) which implies that min_𝑥∈Ω𝑢_𝑗⟶∞as𝑗⟶∞. (60) Now we set ℎ^∗₁=liminf_𝑢→∞ℎ₁(𝑢)∈(],𝑘−∞,∞_∗=limsup_𝑢→∞𝑘(𝑢)∈[−∞,∞). (61) We here consider only the case where ℎ^∗₁∈(−∞,∞) and 𝑘^∗∈(−∞,∞). Either the case ℎ^∗₂=∞ 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, ℎ^∗₁−𝜀<ℎ₁𝑢_𝑗(𝑥)on𝑘𝑢Ω,_𝑗(𝑥)<𝑘^∗+𝜀onΩ. (62) From a computation using Green's formula, it follows that, for any 𝑗≥𝑗₁, 𝜆₁−𝜆_𝑗𝑓_∞Ω𝑎𝑢_𝑗𝜑₁𝑑𝑥≥𝜆_𝑗Ω𝑎ℎ₁𝑢_𝑗𝜑₁𝑑𝑥−_𝜕Ω𝑏𝑘𝑢_𝑗𝜑₁>𝜆𝑑𝜎₁𝑓_∞ℎ^∗₁−𝜀_Ω𝑎𝜑₁𝑑𝑥−𝑘^∗+𝜀_𝜕Ω𝑏𝜑₁𝑑𝜎. (63) As an application of Green's formula, it follows that _Ω𝑎𝜑₁1𝑑𝑥=𝜆_1𝜕Ω𝑏𝑔_∞𝜑₁𝑑𝜎. (64) From these two assertions combined, we obtain that, for any 𝑗≥𝑗₁, 𝜆₁−𝜆_𝑗𝑓_∞Ω𝑎𝑢_𝑗𝜑₁>𝜆𝑑𝑥₁ℎ^∗₁−𝜀𝑔_∞𝑓_∞𝜆₁−𝑘^∗+𝜀_𝜕Ω𝑏𝜑₁𝑑𝜎. (65) On the right-hand side, we see from (7) that lim_𝜀→0𝜆₁ℎ^∗₁−𝜀𝑔_∞𝑓_∞𝜆₁−𝑘^∗=ℎ+𝜀^∗1𝑔_∞𝑓_∞−𝑘^∗>0. (66) This means that, for any 𝑗 large enough, 𝜆₁−𝜆_𝑗𝑓_∞Ω𝑎𝑢_𝑗𝜑₁𝑑𝑥>0, (67) which contradicts the assumption 𝜆_𝑗≥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).