Global Bifurcation of Positive Solutions of Asymptotically Linear Elliptic Problems

and Applied Analysis 3 2. Unilateral Global Bifurcation Theorem of López-Gómez Let U be a Banach space with the normal ‖ ⋅ ‖. Let L(U) stand for the space of linear continuous operators in U. Let J = (a, b) ⊂ R. Let F : R × U → U be a nonlinear operator of the form F (λ, u) = IU − λK +N (λ, u) , (17) whereN : J × U → U is a continuous operator compact on bounded sets such that N (λ, u) = ∘ (‖u‖) , (18) as u → 0 uniformly in any compact interval of J, K ∈ L(U) is a linear compact operator, and r 0 ̸ = 0 is a simple characteristic value ofK; that is, ker [I U − r 0 K] = span [φ 0 ] , (19) for some φ 0 ∈ U \ {0} satisfying φ 0 ∉ R [I U − r 0 K] . (20) Let S be the closure of the set {(λ, u) | (λ, u) ∈ J × U,F (λ, u) = 0, u ̸ = 0} . (21) Let C (resp., C) be the component of S that meets (r 0 , 0) and around (r 0 , 0) lies in S \ Q ε,η (resp., S \ Q ε,η ); see [12, Section 6.4] for the details. Let Σ := {λ ∈ J : dim ker [I U − λK] ≥ 1} . (22) Then we present the unilateral global bifurcation theorem of López-Gómez; see [12, Theorem 6.4.3]. Lemma 7 (see [12], unilateral global bifurcation of López-Gómez). Assume Σ is discrete, r 0 ∈ Σ satisfies (19), and the index Ind(0,N(λ)) changes sign as λ crosses r 0 . Then, for ] ∈ {+, −}, the component C satisfies one of the following: (i) C is unbounded in R × U; (ii) there exists r 1 ∈ Σ \ {r 0 } such that (r 1 , 0) ∈ C; (iii) C contains a point (λ, y) ∈ R × (Y \ {0}) , (23) where Y is the complement of ker[I U − r 0 K] in U. 3. Reduction to a Compact Operator Equation To establish Theorem 1 we begin with the reduction of (1) to a suitable equation for compact operators. According to Gilbarg and Trudinger [13], letK ∞ : C θ (Ω) → C 2+θ (Ω) be the resolvent of the linear boundary value problem −Δu = φ in Ω, ∂u ∂n + b (x) g∞u = 0 on ∂Ω. (24) By Amann [14, Theorem 4.2], K ∞ is uniquely extended to a linear mapping of C(Ω) compactly into C(Ω) and it is strongly positive, meaning that K ∞ φ > 0 on Ω for any φ ∈ C(Ω) with the condition that φ ≥ 0 and φ ̸ ≡ 0 onΩ. Let R ∞ : C 1+θ (∂Ω) → C 2+θ (Ω) be the resolvent of the linear boundary value problem −Δu = 0 in Ω, ∂u ∂n + b (x) g∞u = ψ on ∂Ω. (25) According to Amann [15, 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 = λK ∞ [af (u)] +R∞ [bτ (−k (u))] in C (Ω) . (26) Here τ : C(Ω) → C(∂Ω) is the usual trace operator. Similarly, letK 0 : C θ (Ω) → C 2+θ (Ω) be the resolvent of the linear boundary value problem −Δu = φ in Ω, ∂u ∂n + b (x) g0u = 0 on ∂Ω. (27) Then K 0 is uniquely extended to a linear mapping of C(Ω) compactly into C(Ω) and it is strongly positive. Let R 0 : C 1+θ (∂Ω) → C 2+θ (Ω) be the resolvent of the linear boundary value problem −Δu = 0 in Ω, ∂u ∂n + b (x) g0u = ψ on ∂Ω. (28) ThenR 0 is uniquely extended to a linear mapping of C(∂Ω) compactly into C(Ω). Furthermore, (1) is equivalent to the operator equation u = λK 0 [af (u)] +R0 [bτ (−ζ (u))] in C (Ω) . (29) 4. The Proof of Main Results Obviously, (H1) and (H2) imply that

We make the following assumptions.
(H1)  ∈  1 (R, R) is an odd function with () > 0 for  > 0 and there exist constants  0 ,  ∞ ∈ (0, ∞) and functions , ℎ ∈  1 (R, R), such that A solution  ∈  2 (Ω) of ( 1) is said to be positive if  > 0 on Ω.The purpose of this paper is to study the global bifurcation of positive solutions for the asymptotically linear elliptic eigenvalue problems (1).
Let  = (Ω) be the space of continuous functions on Ω.Then, it is a Banach space with the norm Let Then  is a cone which is normal and has a nonempty interior and  =  − .Moreover, int  = { ∈  :  () > 0 for  ∈ Ω} .
By a constant  ∞ 1 we denote the first eigenvalue of the eigenvalue problem By a constant  0 1 we denote the first eigenvalue of the eigenvalue problem It is well known (cf.Krasnosel'skii [1]) that, for ] ∈ {0, ∞},  ] 1 is positive and simple and that it is a unique eigenvalue with positive eigenfunctions  ] Theorem 1.Let (H0)-(H2) hold.Then, there exists an unbounded, closed, and connected component
Remark 3. Ambrosetti et al. [2] and Umezu [3,4] only studied the bifurcation from infinity for nonlinear elliptic eigenvalue problems.Nonlinear eigenvalue problems of ordinary differential equations have been extensively studied by many authors via fixed point theorem in cones and bifurcations techniques; see Henderson and Wang [5] and Ma [6,7] and the references therein.Ma and Thompson [7] considered the two-point boundary value problem By using the well-known Rabinowitz global bifurcation theorem [8], they proved the following.
Obviously, Corollary 2 is a higher dimensional analogue of Ma and Thompson [7,Theorem 1.1] with  = 1.
Remark 4. Shi [9] studied the exact number of all nontrivial solutions for for  in certain parameter range.He proved the existence of global smooth branches of positive solutions by using the implicit function theorem under some further restrictions on .
Remark 5. Nonlinear elliptic eigenvalue problems have been studied in [4,10] via topological degree and global bifurcation techniques.The positone case (0) ≡ 0 is considered in [10], which is extended to the semipositone case (0) < 0 in [4].An emphasis is, in Theorem 1 and Corollary 2, no assumption imposed on the boundedness of the function ℎ in (3).
Remark 6. Precup [11] applied the Moser-Harnack inequality for nonnegative superharmonic functions to produce a suitable cone and developed fixed point theorem in cones of Krasnoselskii-type to discuss the existence and multiplicity of positive solutions to elliptic boundary value problems The constant  in [11, (3.1)] and the constant  in [11, (3.2)] are not optimal so that [11, Theorem 3.1] is not sharp.However, (10) and (11) in Corollary 2 are optimal.In fact, for the function which satisfies has no positive solution.
The rest of this paper is organized as follows.The proof of our main results is based upon the unilateral global bifurcation theorem of López-Gómez, which is different from the topological degree arguments used in [2][3][4]10].So, in Section 2, we state a preliminary result based upon unilateral global bifurcation theorem of López-Gómez.In Section 3, we reduce (1) into a compact operator equation.Section 4 is devoted to the proof of Theorem 1.

Unilateral Global Bifurcation Theorem of López-Gómez
Let  be a Banach space with the normal ‖ ⋅ ‖.Let L() stand for the space of linear continuous operators in .Let  = (, ) ⊂ R. Let F :  ×  →  be a nonlinear operator of the form where N :  ×  →  is a continuous operator compact on bounded sets such that as  → 0 uniformly in any compact interval of ,  ∈ L() is a linear compact operator, and  0 ̸ = 0 is a simple characteristic value of ; that is, for some  0 ∈  \ {0} satisfying Let S be the closure of the set Let C + (resp., C − ) be the component of S that meets ( 0 , 0) and around ( 0 , 0) lies in S \  − , (resp., S \  + , ); see [12, Section 6.4] for the details. Let Then we present the unilateral global bifurcation theorem of López-Gómez; see [12,Theorem 6.4.3].

Reduction to a Compact Operator Equation
To establish Theorem 1 we begin with the reduction of (1) to a suitable equation for compact operators.According to Gilbarg and Trudinger [13], let K ∞ :   (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem By Amann [14, Theorem 4.2], K ∞ is uniquely extended to a linear mapping of (Ω) compactly into  1 (Ω) and it is strongly positive, meaning that K ∞  > 0 on Ω for any  ∈ (Ω) with the condition that  ≥ 0 and  ̸ ≡ 0 on Ω.Let R ∞ :  1+ (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem According to Amann [15, Section 4], R ∞ is uniquely extended to a linear mapping of (Ω) compactly into (Ω).By the standard regularity argument, problem ( 1) is equivalent to the operator equation Here  : (Ω) → (Ω) is the usual trace operator.Similarly, let K 0 :   (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem Then K 0 is uniquely extended to a linear mapping of (Ω) compactly into  1 (Ω) and it is strongly positive.Let R 0 :  1+ (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem Then R 0 is uniquely extended to a linear mapping of (Ω) compactly into (Ω).Furthermore, ( 1) is equivalent to the operator equation
We consider as a bifurcation problem from the trivial solution  ≡ 0.
then,  is a strongly positive linear operator on .It is easy to verify that  :  →  is completely continuous.From [14, Theorem , it follows that Define N : [0, ∞) ×  →  by then, we have from (30) that locally and uniformly in .
It is very easy to check that (34) enjoys the structural requirements for applying the unilateral global bifurcation theory of [12, Sections 6.4, 6.5] (by a counter example of Dancer [16], the global unilateral theorem of Rabinowitz [8] is false as stated.So, it cannot be used).As the theorem of Crandall and Rabinowitz [17] is applied to get the local bifurcation to positive solutions from ( 0 1 , 0), the algebraic multiplicity of Esquinas and López-Gómez [18] (see [19,Chapter 4]) equals 1 and, therefore, by [12,Theorem 5.6.2] or [19,Proposition 12.3.1],the local index of 0 as a fixed point of  − K 0 changes sign as  crosses  0 1 .Therefore, it follows from Lemma 7 that there exists the component C + that satisfies one of the following: (ii) there exists  * ∈ (0, ∞) with  * ̸ =  0 1 and  ∈ (Ω) which changes its sign on Ω, such that where  is the complement of ker[  −  0 In what follows, we will show that the above Case (ii) and Case (iii) do not occur.
In fact, if (, ) ∈ C + is a nontrivial solution of (34), then  satisfies the problem that is to say,  satisfies the linear problem where We claim that Suppose, on the contrary, that (, ) ∈ C + with  ̸ = 0 and  ∈ .Then, there exists a sequence {(  ,   )} ⊂ C + ∩ with   > 0 for  ∈ N, such that and consequently Combining this with the fact that (, ) is a nontrivial solution of (34) and using the strong maximum principle [20,Theorem 2.4] and (42), it concludes that contracting  ∈ .