Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 105, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.105 GLOBAL BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS INVOLVING NONLOCAL TERMS QINGBO LIU, LAN ZHAO Abstract. We investigate global bifurcation phenomenon for a class of semilinear eigenvalue problems involving nonlocal terms. Under certain assumptions, we demonstrate the existence of a global continuum emanating from the first eigenvalue of the unperturbed problem. As an application of this result, we identify the parameter interval for which positive solutions exist in the problem with general nonlinearities f , where f exhibits asymptotic (q − 1)-linear behavior both near zero and at infinity. To study the global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a semilinear eigenvalue problem. 1. Introduction In this article, we study the global bifurcation phenomenon for the problem: −∆u = λ (∫ Ω |u|qdx ) 2 q−1 |u|q−2u+ h(x, u, λ) in Ω, u = 0 on ∂Ω. (1.1) where Ω is a bounded domain in Rn, n ≥ 2, with a smooth boundary ∂Ω, and λ is a bifurcation parameter. The exponent q satisfies 1 ≤ q ≤ 2. The function h : Ω × R × R satisfies the Carathéodory condition in the first two variable. Problem (1.1) is related to the so-called semilinear eigenvalue problem of the Dirichlet Laplacian −∆u = λ (∫ Ω |u|qdx ) 2 q−1 |u|q−2u in Ω, u = 0 on ∂Ω, (1.2) where the exponent q ∈ [1, 2∗). The first eigenvalue of semilinear problem (1.2), λ1,q(Ω) = inf { ∫ Ω |∇u|2 dx( ∫ Ω |u|q dx )2/q : u ∈W 1,2 0 (Ω), u ̸≡ 0 } , is significant and has been extensively studies in relation to the geometry and function theory of Ω, as well as in the context of mathematical physics issues encountered in engineering. In particular, it is associated with the sharp constant in the Sobolev embedding W 1,2 0 (Ω) ↪→ Lq(Ω): Sq(Ω) = 1√ λ1,q(Ω) = sup { ( ∫ Ω |u|q dx )1/q( ∫ Ω |∇u|2 dx )1/2 : u ∈W 1,2 0 (Ω), u ̸≡ 0 } . S1(Ω) = λ1,1(Ω) is usually referred to as the torsional rigidity of the set Ω, which is a key parameter that measures the ability of rod-like structures (such as beams, shafts, or columns) to resist torsional deformation within the frameworks of elasticity theory and structural engineering. For a given area, the disk (or ball) maximizes S1(Ω) (Pólya-Szegö theorem). λ1,2(Ω) is the principal frequency of the membrane(more generally, the bottom of the spectrum of the Laplacian). For a given area, the disk (or ball) minimizes λ1,2(Ω)(Faber-Krahn inequality). In accordance with 2020 Mathematics Subject Classification. 35B20, 35B32, 35P30. Key words and phrases. Bifurcation method; nonlocal problem; semilinear eigenvalue problem. ©2025. This work is licensed under a CC BY 4.0 license. Submitted August 11, 2025. Published November 11, 2025. 1 2 Q. LIU, L. ZHAO EJDE-2025/105 equation (1.2), λ1,q(Ω) interpolates between the torsional rigidity of a domain and its principal frequency as q ranges from 1 to 2 (see [3]). Rabinowitz [9] studied problem (1.1) with q = 2 by using the topological degree argument, to be precise, the Leray-Schauder degree. He proved the existence of two distinct continua of nontrivial solutions, positive and negative ones, that bifurcate at the point (λ1,2, 0). In this article, we hope to obtain analogous results. Unfortunately, problem (1.1) is not linear, so Leray-Schauder degree argument does not work directly. Upon recognizing that problem (1.1) is homogeneous, and drawing inspiration from the work of Petr Girg and Peter Takáč [7] and Pavel Drábek [5], we employ the Browder-Petryshyn degree instead of the Leray-Schauder degree to address this challenge. Another distinguishing feature of problem (1.1) is that the first equation contains a nonlocal term ∫ Ω |u|qdx and hence the equation is no longer a pointwise identity. This also raises some essential difficulties to study this kind of problems. To tackle the nonlocal term, we partially employ the approach outlined in [4], which investigates a comparable nonlocal equation, namely a Kirchhoff-type equation. It is crucial to highlight that the nonlocal term in [4] is formulated using ∫ Ω |∇u|2 dx, which differs from our specific formulation. Therefore, our methodology does not represent a full replication of the techniques described in [4]. Finally, we establish analogous results as discussed in [7, 9]. We believe that problem (1.1) has not been considered earlier by bifurcation arguments, thus our results with some innovative features are extension of those existed results. In section 2, we investigate the first eigenvalue of eigenvalue problem (1.2) and give its basic properties. This is also a contribution of this article. The remaining section is devoted to some a priori estimates about convergence analysis and the equivalence of norms near (λ1, 0) and (λ1,∞). This part is essential for bifurcation analysis. In section 3, we establish the global bifurcation result for (1.1). Let λ1 ≡ λ1,q denote the first eigenvalue of (1.2). Assume that (H1) h : Ω× R× R satisfies the Carathéodory condition in the first two variable and is locally Hölder continuous in the second variable. There exists a constant C ∈ (0,∞) and p ∈ (2, 2∗) such that |h(x, u;λ)| ≤ C|u|p−1 for a.e. x ∈ Ω and all (u, λ) ∈ R× R. (H2) (for bifurcations from zero) lim s→0 h(x, s, λ) s = 0 uniformly for almost every x ∈ Ω and λ on bounded sets. (H3) (for bifurcations from infinity) lim s→+∞ h(x, s, λ) s = 0 uniformly for almost every x ∈ Ω and λ on bounded sets. Let us point out that the advantage of working with the class locally Hölder continuous is that we can get classical solutions of elliptic equations we deal with by means of Schauder estimates, not merely weak solutions. The first main result of this article is the following theorem. Theorem 1.1. Let q ∈ [1, 2] and h satisfy (H1), (H2). Then the pair (λ1, 0) is a bifurcation point of (1.1). Moreover, there is a component C of the set of nontrivial solutions of (1.1) in R×W 1,2 0 (Ω) whose closure contains (λ1, 0) and it is either unbounded or contains a pair (λ̄, 0) for some λ̄, an eigenvalue of (1.2) with λ̄ ̸= λ1. Let us reformulate Theorem 1.1 in terms of bifurcation from infinity for problem (1.1) at (λ1,∞). Then the second main result of this paper is stated below. We only state the result for bifurcation from infinity without providing a proof, as the proof can be obtained by combining Theorem 1.1 and [10, Theorem 1.6]. EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 3 Theorem 1.2. Let q ∈ [1, 2] and h satisfies (H1), (H3). Then the pair (λ1,∞) is a bifurcation point of (1.1). Moreover, there is a component D of the set of nontrivial solutions of (1.1) in R×W 1,2 0 (Ω) which meets (λ1,∞). If Λ ⊂ R is an interval such that Λ ∩ r(L) = {λ1} where r(L) is the set of real eigenvalue values of (1.2) and M is a neighborhood of (λ1,∞) whose projection on R lies in Λ and whose projection on W 1,2 0 (Ω) is bounded away from 0, then either (i) D − M is bounded in R×W 1,2 0 (Ω) in which case D − M meets R = {(λ, 0) | λ ∈ R} or (ii) D − M is unbounded. If (ii) occurs and D − M has a bounded projection on R, then D − M meets (λ̂,∞) where λ1 ̸= λ̂ ∈ r(L). In Section 4, as a extension and application of the results in Section 3, we consider the problem −∆u = λ (∫ Ω |u|qdx ) 2 q−1 f(u) in Ω, u = 0 on ∂Ω. (1.3) We assume that f satisfies the following conditions: (H4) f : R+ = [0,+∞) → R+ is a locally Hölder continuous function. There exists a positive constant τ such that f(τ) = 0, f(s)s > 0 for s ∈ (0, τ) ∪ (τ,+∞) and there exists a constant κ > 0 such that lim s→τ− f(s) τ − s = κ. (H5) There exists f0 ∈ (0,+∞) such that lim s→0+ f(s) |s|q−2s = f0. (H6) There exists f∞ ∈ (0,+∞) such that lim s→+∞ f(s) |s|q−2s = f∞. The third main result reads as follows. Theorem 1.3. Assume that q ∈ [1, 2]. f satisfies (H4)–(H6) and f0 ̸= f∞. Then (i) if λ ∈ ( min{λ1/f∞, λ1/f0},max{λ1/f∞, λ1/f0} ] , then (1.3) has at least one positive so- lution; (i) if λ ∈ ( max{λ1/f∞, λ1/f0},+∞ ) , then (1.3) has at least two positive solutions. See Figure 1, Figure 1. Schematic diagram of f0, f∞ ∈ (0,+∞) and f0 ̸= f∞. Theorem 1.3 cannot be derived directly. In fact, Theorem 1.1 is not used in the derivation of Theorem 1.3. By expanding f(u) at both zero and infinity, it becomes evident that the pertur- bation of Equation (1.3) does contain nonlocal terms, whereas the perturbation of Equation (1.1) 4 Q. LIU, L. ZHAO EJDE-2025/105 does not include nonlocal terms. Consequently, in Section 4, we will formulate a new bifurcation theorem that is related to (1.3). In the appendix, we give some preliminary results concerning Browder-Petryshyn degree for perturbations of monotone operators. The definition and basic properties can be found in [11, 12]. We now introduce some notation conventions which will be used later in this paper. Let X be the usual Sobolev space W 1,2 0 (Ω) with the norm ∥u∥X = (∫ Ω |∇u|2 )1/2 and X ′ be its dual space. Sometimes, we omit dx in the integral symbol. Denote by ⟨·, ·⟩ the duality pairing between X and X ′. We write un ⇀ u and un → u for the weak and strong convergence of the sequence {un} in X, respectively. Let Lp be the usual Lebesgue space with the norm ∥u∥p = (∫ Ω |u|pdx )1/p . For a measurable set A of Rn we denote its measure by |A|. Also, we denote by c, ci, C, and Ci, i ∈ N, general positive constants the exact value may be different from line to line. The rest of this paper is arranged as follows. In Section 2, we give some preliminaries which will be used later in this paper. In Section 3, we give the proof of Theorem 1.1. In Section 4, we give the proof of Theorem 1.3. 2. Preliminaries 2.1. Properties of the first eigenvalue λ1. By the compactness of the embedding W 1,2 0 (Ω) ↪→ Lp(Ω), there exists a minimizer of λ1 and λ1 is well defined. We are going to study the properties of λ1 = inf { ∫ Ω |∇u|2 dx( ∫ Ω |u|q dx )2/q : u ∈W 1,2 0 (Ω), u ̸≡ 0 } . These properties are important in the study of global bifurcation phenomena. Lemma 2.1. Let λ1 is the first eigenvalue of (1.2) and φ1 is an eigenfunction corresponding to λ1. Then φ1 ∈ C1,α(Ω̄) for some α ∈ (0, 1) and ∂φ1/∂ν < 0 if φ1 is nonnegative, where ν is the outer unit normal at x ∈ ∂Ω. Proof. In fact, u belongs C2(Ω). The method of proof is so-called the bootstrap argument, we refer readers to [2, Theorem 1.16]. Furthermore, if φ1 ≥ 0, by the strong maximum principle(see [8]), ∂φ1/∂ν < 0 for all x ∈ ∂Ω. □ Lemma 2.2. Let φ1 be an eigenfunction associated with λ1, then either φ1 > 0 or φ1 < 0 in Ω, i.e. λ1 is the principal eigenvalue of (1.2). Proof. We notice that if φ1 is an eigenfunction, so is v := |φ1|. Without loss of generality, we shall assume that ∥v∥q = 1. So we have −∆v = λ1v q−1 in Ω, v = 0 on ∂Ω. By the strong maximum principle [8], we know that v > 0 in the whole domain. By the continuity of φ1, either φ1 or −φ1 is positive in the whole domain. □ Remark 2.3. Eigenfunction φ1 can be normalized by φ1 > 0 in Ω and ∫ Ω φq 1 = 1. Lemma 2.4. λ1 is simple. Proof. We only discuss q ∈ [1, 2) and q = 2 is easy. Consider the auxiliary problem −∆u = uq−1 in Ω, u = 0 on ∂Ω. (2.1) Reference [1, Theorem 8.4.1] states that the positive solution of (2.1) is unique with q ∈ [1, 2). Let u, v be two eigenfunctions associated with λ1. Then λ2−q 1 u ∥u∥1/qq and λ2−q 1 v ∥v∥1/qq EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 5 are both solutions to (2.1). Hence u = (∥u∥q ∥v∥q )1/q v, which shows that λ1 is simple. □ Lemma 2.5. (1.2) has a positive solution if and only if λ = λ1. Proof. Suppose on the contrary that (1.2) with λ > λ1 has a positive solution v, and let u be a positive eigenfunction corresponding to λ1. There exists a constant t > 0 large enough such that tv ≥ u. Clearly, ṽ := tv is also an eigenfunction of (1.2). Define Φ and Ψ on X by Φ(w) = 1 2 ∥w∥2X , Ψ(w) = 1 2 (∫ Ω wqdx )2/q . Then the energy functional corresponding to (1.2) is J(w) = (Φ− λΨ)(w) = 1 2 ∥w∥2X − λ 2 (∫ Ω wqdx )2/q . For all φ ∈ C∞ c (Ω), let Φ′(w)φ = ∫ Ω ∇w∇φ, Ψ′(w)φ = (∫ Ω wqdx ) 2 q−1 ∫ Ω wq−1φ. Then w is a weak solution of (1.2) if and only if Φ′(w) = λΨ′(w). For φ ≥ 0, we have ⟨Ψ′(u), φ⟩ ≤ ⟨Ψ′(ṽ), φ⟩. Then ⟨Φ′(u), φ⟩ = ⟨λ1Ψ′(u), φ⟩ ≤ ⟨λ1Ψ′(ṽ), φ⟩ = ⟨λΨ′(ηṽ), φ⟩ = ⟨Φ′(ηṽ), φ⟩, where η = λ1/λ < 1. Taking φ = (u−ηṽ)+ as a test function in ⟨Φ′(u), φ⟩ ≤ ⟨Φ′(ηṽ), φ⟩, it follows that ∇(u− ηṽ)+ = 0, this implies (u− ηṽ)+ = 0 and so u ≤ ηṽ in Ω. Repeating this argument n times, we obtain that 0 ≤ u ≤ ηnṽ. Letting n → +∞, we obtain u ≡ 0. This is a contradiction. So v must change sign. □ Lemma 2.6. λ1 is isolated. Proof. Let v be any eigenfunction associated to an eigenvalue λ > λ1 and N be its any nodal domain. Then we have v|N ∈W 1,2 0 (N ). Define w = { v for x ∈ N 0 for x ∈ Ω\N It is easy to see that w ∈ X and we claim |N | ≥ C(λ), where C(λ) is a constant and is only related to λ. We only consider the case n ≥ 3, and n = 2 is simple. We have∫ N |∇w|2dx = λ (∫ N |v|qdx ) 2 q−1 ∫ N wqdx. By the Hölder inequality and the Sobolev embeddings we have c∥w∥22∗ ≤ ∫ N |∇w|2dx ≤ λ (∫ N |w|qdx )2/q ≤ λ∥w∥22∗ |N |(1− q 2∗ ) 2 q . where c > 0 is related to the best embedding constant. Based on the aforementioned fact, we substantiate the assertion: |N | ≥ ( c λ ) 2∗ 2∗−q q 2 ≡ C(λ). Now we prove Lemma 2.6 by contradiction. Assume that there exists a sequence of eigenvalues λn ∈ (λ1, δ) for some constant δ > λ1 which converges to λ1. Let un be the corresponding eigenfunctions. Lemma 2.5 implies that un changes sign. Integration by parts gives∫ Ω |∇un|2dx = λn (∫ Ω |un|qdx )2/q . 6 Q. LIU, L. ZHAO EJDE-2025/105 We define vn := un ∥un∥q . Obviously, vn is bounded in X so there exists a subsequence, denoted again by vn, and v ∈ X such that vn ⇀ v in X and vn → v in Lq(Ω). Norm ∥ · ∥X is sequentially weakly lower semi-continuous, so we have that ∫ Ω |∇v|2dx ≤ lim inf n→+∞ ∫ Ω |∇vn|2dx = lim inf n→+∞ λn = λ1. On the other hand, ∥vn∥q = 1 and vn → v in Lq(Ω) imply that ∥v∥q = 1. It follows that∫ Ω |∇v|2dx ≤ λ1 (∫ Ω vqdx )2/q . The above inequality and the variational characterization of λ1 imply that∫ Ω |∇v|2dx = λ1 (∫ Ω vqdx )2/q . Without loss of generality, we may assume that v > 0 in Ω. Since vn ⇀ v in X, passing if necessary to a subsequence, we can assume that vn → v a.e. in Ω. Therefore, we arrive at the conclusion that |B− n | → 0, where B− n denotes the negative set of un. This presents a contradiction to the aforementioned assertion. □ Because of Lemma 2.6, we have shown that λ1 is an isolated eigenvalue of (1.2); i.e. if we let λ2 = inf{λ > λ1 : λ is an eigenvalue of problem (1.2)}, then λ1 < λ2. Moreover, we have the following conclusion. Proposition 2.7. There exists δ > 0 such that for all q ∈ [1, 2], there is no eigenvalue of problem (1.2) in (λ1, λ1 + δ]. 2.2. Equivalence of norms near (λ1, 0) and (λ1,∞). The following lemma is a useful conse- quence of hypothesis (H2) or (H3). Lemma 2.8. Let u ∈ L∞(Ω) and u ̸≡ 0 in Ω. (i) If (H2) is satisfied, then h(x, u, λ) ∥u∥L∞(Ω) → 0 as ∥u∥L∞(Ω) → 0 holds for a.e. x ∈ Ω and uniformly for every λ ∈ R. (ii) If (H3) is satisfied, then h(x, u, λ) ∥u∥L∞(Ω) → 0 as ∥u∥L∞(Ω) → ∞ holds for a.e. x ∈ Ω and uniformly for every λ ∈ R. Proof. We first notice that h(x, u, λ) = 0 if u(x) = 0, and estimate |h(x, un, λ)| ∥un∥L∞(Ω) = |h(x, un, λ)| |un(x)| |un(x)| ∥un∥L∞(Ω) ≤ |h(x, un, λ)| |un(x)| if un(x) ̸= 0. From ∥un∥L∞(Ω) → 0 we obtain un(x) → 0 uniformly for a.e. x ∈ Ω. This gives (i). To prove (ii), we split the domain as Ω =Mn ∪Nn where Mn = { x ∈ Ω : |un(x)| ≤ ∥un∥ 1 2(p−1) L∞(Ω) } , Nn = { x ∈ Ω : |un(x)| > ∥un∥ 1 2(p−1) L∞(Ω) } . For x ∈Mn we infer that |h(un)| ∥un∥L∞(Ω) = |h(un)| |un|p−1 |un|p−1 ∥un∥L∞(Ω) ≤ C ∥un∥1/2L∞(Ω) ∥un∥L∞(Ω) . EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 7 For x ∈ Nn we infer that |h(un)| ∥un∥L∞(Ω) = |h(un)| |un| |un| ∥un∥L∞(Ω) ≤ |h(un)| |un| . Now let χMn and χNn denote the characteristic functions of the sets Mn and Nn, respectively. By combining the previous inequalities, we obtain |h(un)| ∥un∥L∞(Ω) ≤ C ∥un∥1/2L∞(Ω) ∥un∥L∞(Ω) χMn(x) + |h(un)| |un| χNn(x). It is easy to see that h(un) ∥un∥L∞(Ω) → 0 as ∥un∥L∞(Ω) → +∞ holds for a.e. x ∈ Ω. □ Corollary 2.9. Let 1 ≤ r <∞, and (H2) or (H3) is satisfied. Then ∥ h(x, u, λ) ∥u∥L∞(Ω) ∥r = (∫ Ω ∣∣ h(x, u, λ) ∥u∥L∞(Ω) ∣∣r)1/r → 0 as ∥u∥L∞(Ω) → 0 or ∥u∥L∞(Ω) → ∞ uniformly for every λ ∈ R. Let us consider a sequence of nontrivial solutions {(λn, un)}∞n=1 of problem (1.1), i.e., for each n = 1, 2, . . ., the integral identity∫ Ω ∇un∇ϕdx = λn (∫ Ω |un|qdx ) 2 q−1 ∫ Ω |un|q−2unϕdx+ ∫ Ω h(x, un;λ)ϕdx holds for all ϕ ∈W 1,2 0 (Ω). We assume that 0 < λn ≤ λ2 − δ, n = 1, 2, . . . , where δ ∈ (λ2 − λ1) is a constant. Because of [2, Theorem 1.16], un ∈ C2(Ω). More over we have the following result. Theorem 2.10. Let {(λn, un)}∞n=1 be as above. Then the following three statements are equivalent, as n→ ∞, (i) ∥un∥W 1,2 0 (Ω) → 0, (ii) ∥un∥L∞(Ω) → 0, (iii) ∥un∥C1,β(Ω) → 0. Proof. Clearly, (iii) implies (i) and (ii). (i) ⇒ (ii): We denote wn = un/∥un∥X satisfying ∥wn∥X = 1 and −∆wn = λn (∫ Ω |wn|qdx ) 2 q−1 |wn|q−2wn + h(x,wn∥un∥X , λn) ∥un∥X . Based on the hypotheses 1 ≤ q ≤ 2 and (H1), the right-hand side, represented as f(x,wn), satisfies f(x,wn) = λn∥wn∥2−q q |wn|q−2wn + h (x,wn∥un∥X , λn) ∥un∥X ≤ C1|wn|q−1 + C2∥un∥p−2 X |wn|p−1 (2.2) where C1 and C2 are constants that are independent of ∥un∥X . Our aim is to obtain a priori estimates and determine the L∞-norm of wn, which is controlled by the W 1,2 0 -norm of wn. How- ever, it is important to recognize that the coefficient of |wn|p−1 is associated with the value of ∥un∥p−2 X , which is variable! Consequently, a comprehensive analysis utilizing a bootstrap argument is required. Now, we proceed as in the proof of [2, Theorem 1.16]. Step 1. By the Sobolev embedding theorem it follows that wn ∈ L2∗ and satisfies ∥wn∥2∗ ≤ C∥wn∥X = C (2.3) where C is a constant that is independent of ∥un∥X . 8 Q. LIU, L. ZHAO EJDE-2025/105 Step 2. Based on the information provided in (2.2) and the condition p > q, it follows that f(x,wn) ∈ Lr with r = 2∗ p−1 and ∥f(x,wn)∥rr ≤ ∫ Ω ∣∣C1|wn|q−1 + C2∥un∥p−2 X |wn|p−1 ∣∣rdx ≤ C1 ∫ Ω |wn|(q−1)rdx+ C2∥un∥(p−2)r X ∫ Ω |wn|(p−1)rdx = C1∥wn∥(q−1)r (q−1)r + C2∥un∥(p−2)r X ∥wn∥(p−1)r (p−1)r ≤ C1∥wn∥(q−1)r 2∗ + C2∥un∥(p−2)r X ∥wn∥(p−1)r 2∗ . Hence, we derive that ∥f(x,wn)∥r ≤ C1∥wn∥q−1 2∗ + C2∥un∥p−2 X ∥wn∥p−1 2∗ (2.4) where C1 and C2 are constants that are independent of ∥un∥X . Step 3. Since −∆wn = f(x,wn), L p-estimates yields u ∈W 2,r and ∥wn∥W 2,r(Ω) ≤ C∥f(x,wn)∥r (2.5) where C is a constant that is independent of ∥un∥X . If 2r > n then u ∈ C0,γ(Ω). Otherwise, we can repeat Steps 1–3. After a finite number of times, one finally finds a number r∗ such that u ∈ W 2,r∗ with 2r∗ > n. Then the Sobolev embedding theorem yields again W 2,r∗(Ω) ↪→ C0,γ(Ω) with γ ≤ 1 and ∥wn∥C0,γ(Ω) ≤ C∥wn∥W 2,r(Ω) (2.6) where C is a constant that is independent of ∥un∥X . From (2.3)-(2.6), we obtain ∥wn∥C0,γ(Ω) ≤ C1 + C2∥un∥p−2 X (2.7) where C1 and C2 are constants that are independent of ∥un∥X . Furthermore, based on Schauder estimates, we obtain wn ∈ C2,γ(Ω) and ∥wn∥C2,γ(Ω) ≤ C1 + C2∥un∥p−2 X (2.8) where C1 and C2 are constants that are independent of ∥un∥X . (2.8) implies that ∥un∥∞ ≤ C∥un∥C2,γ(Ω) ≤ ( C1 + C2∥un∥p−2 X ) ∥un∥X . (2.9) However, because p− 2 > 0, (i) ⇒ (ii) is proved. (ii) ⇒ (iii). proceeding as above, we obtain ∥un∥C1,β(Ω) ≤ C∥un∥C2,γ(Ω) ≤ ( C1 + C2∥un∥p−2 ∞ ) ∥un∥∞ (2.10) where C1 and C2 are constants that are independent of ∥un∥∞. □ Corollary 2.11. Let {(λn, un)}∞n=1 be as above with ∥un∥∞ → 0. Then there exists subsequence, which we denote again by λn and un, such that as n→ ∞ for some α ∈ (0, 1), λn → λ1, un ∥un∥∞ → ± φ1 ∥φ1∥∞ in C1,α(Ω) (2.11) where λ1 is the first eigenvalue of (1.2) and φ1 is an eigenfunction corresponding λ1. Proof. We denote wn = un/∥un∥∞ which satisfies∫ Ω ∇wn∇ϕ = λn (∫ Ω |wn|q ) 2 q−1 ∫ Ω |wn|q−2wnϕ+ ∫ Ω h(x,wn∥un∥∞, λn) ∥un∥∞ ϕ (2.12) for all ϕ ∈ X. Using the compact embedding C1,β ↪→ C1,α with 0 < α < β < 1, the sequence wn contains a subsequence that converges in C1,α to some w; we denote it again by wn → w. We let n→ ∞ in (2.12) and use Corollary 2.9, to conclude that w ∈ C1,α must satisfy∫ Ω ∇w∇ϕdx = λ∗ (∫ Ω |w|q dx ) 2 q−1 ∫ Ω |w|q−2wϕdx (2.13) EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 9 where λ∗ = limn→∞ λn. Since 0 ≤ λ∗ ≤ λ2 − δ and λ1 is the only eigenvalue of (1.2) in the open interval (0, λ2), we must have λ∗ = λ1. In addition, λ1 being a simple eigenvalue, we have w = kφ1 in Ω where k satisfies |k| · ∥φ1∥∞ = ∥w∥∞ = ∥wn∥∞ = 1. This gives (2.11). □ Comparable results can be achieved concerning Theorem 2.10 and Corollary 2.11 in the context of (λ1,∞). Theorem 2.12. Let {(λn, un)}∞n=1 be as above. Then the following three statements are equivalent, as n→ ∞, (i) ∥un∥W 1,2 0 (Ω) → ∞, (ii) ∥un∥L∞(Ω) → ∞, (iii) ∥un∥C1,β(Ω) → ∞. Corollary 2.13. Let {(λn, un)}∞n=1 be as above with ∥un∥∞ → ∞. Then there exists subsequence, which we denote again by λn and un, such that as n→ ∞ for some α ∈ (0, 1) λn → λ1, un ∥un∥∞ → ± φ1 ∥φ1∥∞ in C1,α(Ω) where λ1 is the first eigenvalue of (1.2) and φ1 is an eigenfunction corresponding λ1. 3. Global bifurcation It is known that u ∈W 1,2 0 (Ω) is a solution of problem (1.1) (in the weak sense) if and only if a pair (λ, u) ∈ R×W 1,2 0 (Ω) that satisfies the integral identity∫ Ω ∇u∇ϕdx = λ (∫ Ω |u|qdx ) 2 q−1 ∫ Ω |u|q−2uϕ dx+ ∫ Ω h(x, u;λ)ϕdx for all ϕ ∈W 1,2 0 (Ω). The last equation is equivalent to the operator equation Φ(u) = λΨ(u) +H(λ, u) with all terms valued in the dual space X ′ = W−1,2′(Ω) of X and the operators Φ,Ψ, H(λ, ·) : X → X ′ defined as follows, for all u, ϕ ∈ X and λ ∈ R: ⟨Φ(u), ϕ⟩X = ∫ Ω ∇u∇ϕdx, ⟨Ψ(u), ϕ⟩X = (∫ Ω |u|qdx ) 2 q−1 ∫ Ω |u|q−2uϕ dx, ⟨H(λ, u), ϕ⟩X = ∫ Ω h(x, u, λ)ϕ(x) dx. It is easy to see that the operator Φ : X → X ′ is continuous, coercive, strictly monotone and satisfies condition (S+). The operator Ψ : X → X ′ can be extended to a continuous operator Ψ̃ : Lq(Ω) → (Lq(Ω))′ = Lq′(Ω) in a unique way. Consequently, Ψ decomposed as Ψ : X ↪→ Lq(Ω) Ψ̃−→ Lq′(Ω) ↪→ X ′ is compact by Rellich’s theorem. Finally, given λ ∈ R, operator H(λ, · ) : X → X ′ is also compact by Rellich’s theorem. Thus Gλ(u) = G(λ, u) ≡ Φ(u)−Ψ(u)−H(λ, u) can be viewed as a compact perturbation of (S+)-type operator Ψ and is also (S+)-type. So its Browder-Petryshyn degree can be defined. Now, we give a result that shows discontinuity for Browder-Petryshyn degree at the first eigen- value λ1. 10 Q. LIU, L. ZHAO EJDE-2025/105 Lemma 3.1. For all r > 0 and all 0 < δ < λ2 − λ1 we have Deg [ Φ− (λ1 ± δ)Ψ;Br(0), 0 ] = ∓1. Proof. Given R > 0 fixed, we define ψ : R+ → R+ as ψ(t) =  0 for 0 ≤ t ≤ R, δ R (t−R) 2 for R < t < 2R, 2δ(t− 2R) + δR for 2R ≤ t <∞. (3.1) Clearly, ψ is continuously differentiable, monotone increasing, and convex on R+ satisfying ψ′(t) =  0 for 0 ≤ t ≤ R, 2δ R (t−R) for R < t < 2R, 2δ for 2R ≤ t <∞. (3.2) Now we consider the functional Tλ : X → R defined as Tλ(u) = 1 2 〈 Φ(u), u 〉 X − λ 2 〈 Ψ(u), u 〉 X + ψ (1 2 〈 Ψ(u), u 〉 X ) . Every critical point u0 ∈ X of Tλ is a solution of the operator equation T ′ λ(u) = Φ(u)− [ λ− ψ′ (1 2 ⟨Ψ(u), u⟩X )] Ψ(u) = 0 inX ′ . (3.3) Next we investigate the above equation. Assuming λ ≤ λ1 + δ < λ2 we have λ− 2δ ≤ λ− ψ′ (1 2 ⟨Ψ(u), u⟩X ) ≤ λ ≤ λ1 + δ < λ2. Therefore, if (3.3) is valid, we have following two cases: (i) u0 = 0, and (ii) λ−ψ′( 12 ⟨Ψ(u0), u0⟩X) = λ1 and u0 = αφ1 for some constant α ∈ R \ {0}. Here φ1 is an eigenfunction corresponding to λ1 satisfies ∥φ1∥2q = 1, and α, because of the homogeneity, satisfies λ− ψ′ ( |α|2 2 ) = λ1. Because α depends on λ, we sometimes write α = αλ. Next, we discuss the value of λ in three cases. Case 1: λ < λ1. We have ψ′ (|α|2/2) = λ− λ1 < 0. Combining with (3.2), the only zero of Tλ is the u = 0 ∈ X; it is the global minimizer for Tλ. We apply Theorem 5.8 to conclude that Deg[Tλ;Br(0), 0] = 1 for all r > 0. (3.4) Case 2: λ = λ1. In this situation, ψ′(|α|2/2) = λ1 − λ1 = 0. Combining with (3.2), we have 0 ≤ |αλ1 |2 2 ≤ R. Combining with (3.1) again, it is easy to see that Tλ1 (αλ1 φ1) = 0. Case 3: λ1 < λ ≤ λ1 + δ. In this situation, ψ′ (|α|2/2) ∈ (0, δ]. Combining this with (3.2), we have R < |αλ|2/2 ≤ 3R 2 . It can be known by direct calculation that Tλ1+δ(±αλ1+δφ1) < 0 = T0. According to (3.3), 0 and ±αλ1+δφ1 are the whole set of zeros of Tλ1+δ. Next, we proof Tλ is coercive on X. In fact, we fix 1 2 ⟨Ψ(u), u⟩X = 2R. This is a C1 manifold and contains origin u = 0. Then we have Tλ(u) = 1 2 ⟨Φ(u), u⟩X − λ 2 ⟨Ψ(u), u⟩X + 2δ (1 2 ⟨Ψ(u), u⟩X − 2R ) + δR = 1 2 ⟨Φ(u), u⟩X − λ− 2δ 2 ⟨Ψ(u), u⟩X − 3δR EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 11 ≥ 1 2 ⟨Φ(u), u⟩X − λ1 − δ 2 ⟨Φ(u), u⟩X − 3δR ≥ 1 2 ( 1− λ1 − δ λ1 ) ⟨Φ(u), u⟩X − 3δR = δ ( 1 2λ1 ⟨Φ(u), u⟩X − 3R ) . Let ∥u∥X → ∞, we obtain Tλ(u) → ∞. Thus, ±αλ1+δφ1 are the global minimizers for Tλ1+δ. By Theorem 5.8 again, we obtain Deg [ T ′ λ1+δ;Bσ(±αλ1+δφ1), 0 ] = 1 for every σ > 0 small enough. (3.5) Without loss of generality, we assume σ < 1 2αλ1+δ∥φ1∥X . Set rδ = 2αλ1+δ∥φ1∥X , then rδ > αλ1+δ∥φ1∥X + σ. Let Tt(u) = tT ′ λ1−δ(u) + (1− t)T ′ λ1+δ(u) t ∈ [0, 1] and ∥u∥X > rδ. Noticing that Tt(u) ̸= 0 holds for all u ∈ ∂Brδ(0) and t ∈ [0, 1] due to choice of σ and rδ, by the homotopy invariance of degree, we obtain Deg [ T ′ λ1+δ;Br(0), 0 ] = Deg [ T ′ λ1−δ;Br(0), 0 ] = 1 for every r > rδ. (3.6) On the other hand, 0 is also an isolated zero of T ′ λ with λ = λ1 + δ and Deg [ T ′ λ1+δ;Bσ′(0), 0 ] is well defined for every σ′ > 0 small enough. Without loss of generality, we still assume σ′ < 1 2αλ1+δ∥φ1∥X . From (3.4), (3.6) and additivity property of the degree we deduce for all r > rδ, Deg [ T ′ λ1+δ;Bσ(αλ1+δφ1), 0 ] +Deg [ T ′ λ1+δ;Bσ(−αλ1+δφ1), 0 ] +Deg [ T ′ λ1+δ;Bσ′(0), 0 ] = Deg [ T ′ λ1+δ;Br(0), 0 ] = Deg [ T ′ λ1−δ;Br(0), 0 ] = 1. From (3.5), we obtain Deg [ T ′ λ1+δ;Bσ′(0), 0 ] = −1. To complete the proof, we only need to prove that there exists σ′ small enough such that 1 2 ⟨Ψ(u), u⟩X < R for ∥u∥X < σ′. This is because, if that is correct, from (3.2) we have Φ(u)− (λ1 + δ)Ψ(u) = Φ(u)− [ (λ1 + δ) + ψ′ (1 2 ⟨Ψ(u), u⟩ )] Ψ(u) = T ′ λ1+δ(u). Then Deg [Φ− (λ1 + δ)Ψ;Bσ′(0), 0] = Deg [ T ′ λ1+δ;Bσ′(0), 0 ] = −1. Indeed that is true. Because, from the definition of λ1, we have λ1⟨Ψ(u), u⟩X ≤ ⟨Φ(u), u⟩ = ∥u∥X . Since u = 0 ∈ X is the only solution to the operator equation Φ(u) = (λ1 + δ)Ψ(u). We obtain Deg [Φ− (λ1 + δ)Ψ;Br(0), 0] = −1 for every r > 0. By analogous arguments, we infer that Deg [Φ− (λ1 − δ)Ψ;Br(0), 0] = 1 for every r > 0. □ Proof of Theorem 1.1. According to [6], if (λ, 0) is a bifurcation point, then λ is an eigenvalue for the nonlinear eigenvalue problem Φ(u) − λΨ(u) = 0. So for each λ ∈ (0, λ2)\{λ1}, u = 0 ∈ X is an isolated solution of Gλ(u) = 0. Thus one can find R > 0 small enough, such that Deg[Gλ1±δ;Br(0), 0] remains constant with respect to r ∈ (0, R). Now, we assert that there exists R′ ∈ (0, R) such that Φ(u)− (λ1 ± δ)Ψ(u)− tH(λ1 ± δ, u) ̸= 0 (3.7) 12 Q. LIU, L. ZHAO EJDE-2025/105 for all u ∈ ∂BR′(0) and t ∈ [0, 1]. If not, for all R′ ∈ (0, R) there exist u ∈ ∂BR′(0) and t ∈ [0, 1] such that Φ(u)− (λ1 ± δ)Ψ(u)− tH(λ1 ± δ, u) = 0. Thus, we can find a sequence {rn}∞n=1 ⊂ (0, R), rn → 0, with {un}∞n=1 ⊂ X, un ∈ ∂Brn(0), and {tn}∞n=1 ⊂ [0, 1], such that above equation holds with un and tn in place of u and t, respectively. So, for all ϕ ∈ C∞ c (Ω), we have∫ Ω ∇un∇ϕ = (λ1 ± δ) (∫ Ω |un|q ) 2 q−1 ∫ Ω |un|q−2un ϕ+ tn ∫ Ω h(x, un, λ)ϕ. (3.8) Since un ∈ ∂Brn(0), we have ∥un∥X → 0. Dividing both sides of the equation by ∥un∥X and denoting un/∥un∥X by wn the sequence {wn} is bounded in X. This means that there exists a w ∈ X and subsequence that we call again wn, such that • wn ⇀ w in X; • wn → w in Lp(Ω), ∀ p ∈ [1, 2∗); • wn(x) → w(x) a.e. in Ω; • there exists v ∈ Lp(Ω) such that |wn(x)| ≤ v(x) a.e. in Ω and for all n. Then we obtain∫ Ω ∇wn∇ϕ = (λ1 ± δ) (∫ Ω |wn|q ) 2 q−1 ∫ Ω |wn|q−2wnϕ+ tn ∫ Ω h(x, un, λ) ∥un∥X ϕ. (3.9) Next, it is sufficient to show that lim n→∞ ∫ Ω h(x, un, λ) ∥un∥X ϕ dx = 0. (3.10) Since combining with weak convergence, Hölder inequality and Lebesgue dominated convergence theorem, we can obtain that ∫ Ω ∇wn∇ϕdx→ ∫ Ω ∇w∇ϕdx,(∫ Ω |wn|qdx ) 2 q−1 ∫ Ω |wn|q−2wnϕdx→ (∫ Ω |w|qdx ) 2 q−1 ∫ Ω |w|q−2wϕdx. This means that there exists 0 ̸= w ∈ X satisfying∫ Ω ∇w∇ϕdx = (λ1 ± δ) (∫ Ω |w|qdx ) 2 q−1 ∫ Ω |w|q−2wϕdx. This contradicts that λ1 ± δ are not eigenvalues. Now we prove (3.10). From Lemma 2.9 and Theorem 2.10, we have∫ Ω h(x, un, λ) ∥un∥X ϕ dx = ∫ Ω h(x, un, λ) ∥un∥∞ ∥un∥∞ ∥un∥X ϕdx ≤ ( C1 + C2∥un∥p−2 X )∫ Ω h(x, un, λ) ∥un∥∞ ϕdx→ 0. Therefore, the degree of the homotopy operator Φ(u)− (λ1 ± δ)Ψ(u)− tH(λ1 ± δ, u), t ∈ [0, 1], is well defined. This operator connects Gλ1±δ with Φ− (λ1 ± δ)Ψ. Consequently, by Lemma 3.1, we have Deg[Gλ1±δ;Br(0), 0] = Deg[Gλ1±δ;BR′(0), 0] = Deg[Φ− (λ1 ± δ)Ψ;BR′(0), 0] = ∓1. Next, we can proceed step by step as in the original proof of Rabinowitz [9]. This concludes the proof. □ □ By Corollary 2.11 and Corollary 2.13, we have the following result. Corollary 3.2. Let C is a component of the set of nontrivial solutions of (1.1) in R × X in Theorem 1.1 and D is a component of the set of nontrivial solutions of (1.1) in R×X in Theorem 1.2. EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 13 (i) If (µn, un) ∈ C and is near (λ1, 0), then un = snφ1 +wn, where wn = o(|sn|) as |sn| → 0 and ∫ Ω wnφ1dx = 0. (ii) If (µn, un) ∈ D and is near (λ1,∞), then un = snφ1+wn, where wn = o(|sn|) as |sn| → ∞ and ∫ Ω wnφ1dx = 0. 4. Positive solutions In this Section, we consider the problem −∆u = λ (∫ Ω |u|qdx ) 2 q−1 f(u) in Ω, u = 0 on ∂Ω. (4.1) Using (H5) we define ξ : R+ → R+ as f(s) = f0|s|q−2s+ ξ(s) with lim s→0+ ξ(s) |s|q−2s = 0. Then (4.1) is transformed into −∆u = λf0 (∫ Ω |u|qdx ) 2 q−1 |u|q−2u+ λ (∫ Ω |u|qdx ) 2 q−1 ξ(u) in Ω, u = 0 on ∂Ω, (4.2) as a bifurcation problem from the trivial solution axis. We intend to utilize Theorem 1.1 in the context of Problem (4.2). However, we observe that Equations (1.1) and (4.2) are not entirely identical; they share a similar structure only in certain aspects. To be precise, it is observed that the second term on the right-hand side of Equation (4.2) includes a nonlocal term, ( ∫ Ω |u|qdx) 2 q−1, whereas the second term on the right-hand side of Equation (1.1) does not incorporate this term. Consequently, it is necessary to formulate a theorem analogous to Theorem 1.1 for Equation (1.1). To ensure comprehensiveness, we generalize (4.2) into the following equation. −∆u = λ (∫ Ω |u|qdx ) 2 q−1 |u|q−2u+ (∫ Ω |u|qdx ) 2 q−1 g(x, u, λ) in Ω, u = 0 on ∂Ω, (4.3) where g : Ω × R × R satisfies the Carathéodory condition in the first two variable and is locally Hölder continuous about the second variable. There exists a constant C ∈ (0,∞) and p ∈ (2, 2∗) such that |g(x, u, λ)| ≤ C|u|p−1 for a.e. x ∈ Ω and all (u, λ) ∈ R × R. Furthermore, we assume that g satisfies the following hypotheses: (H7) (for bifurcations from zero) lim s→0+ g(x, s, λ) sq−1 = 0 uniformly for almost every x ∈ Ω and λ on bounded sets. (H8) (for bifurcations from infinity) lim s→+∞ g(x, s, λ) sq−1 = 0 uniformly for almost every x ∈ Ω and λ on bounded sets. We aim to provide a proof of the subsequent theorem. Theorem 4.1. Let q ∈ [1, 2] and g satisfies (H7). Then the pair (λ1, 0) is a bifurcation point of (4.3). Moreover, there is a component C of the set of nontrivial solutions of (4.3) in R ×X whose closure contains (λ1, 0) and it is either unbounded or contains a pair ( λ, 0 ) for some λ, an eigenvalue of (1.2) with λ ̸= λ1. 14 Q. LIU, L. ZHAO EJDE-2025/105 Upon examining the proof of Theorem 1.1, it becomes evident that the critical component lies in the demonstration of (3.10). However the validity of (3.10) is contingent upon Lemma 2.8 and Theorem 2.10. Consequently, it is imperative to establish analogous conclusions for the function ∥u∥2−q q g(x, u, λ). We shall now continue with this verification. Initially, it is evident that the analogue of Lemma 2.8 can be established through a proof that is analogous to that of Lemma 2.8. This constitutes the subsequent lemma. Lemma 4.2. Let u ∈ L∞(Ω), u ̸≡ 0 in Ω. (i) If hypothesis (H7) is satisfied, then g(x, u, λ) ∥u∥q−1 L∞(Ω) → 0 as ∥u∥L∞(Ω) → 0 holds for a.e. x ∈ Ω and uniformly for every λ ∈ R. (ii) If hypothesis (H8) is satisfied, then g(x, u, λ) ∥u∥q−1 L∞(Ω) → 0 as ∥u∥L∞(Ω) → ∞ holds for a.e. x ∈ Ω and uniformly for every λ ∈ R. Next, we investigate Theorem 2.10, the equivalence of norms. An analysis of the proof of Theorem 2.10 reveals that a pivotal aspect is the establishment of (2.2) and Step 2. When h is replaced by ∥u∥2−q q g(x, u, λ), the resulting expression is as follows f(x,wn) = λn∥wn∥2−q q |wn|q−2wn + ∥un∥2−q q g(x,wn∥un∥X , λn) ∥un∥X ≤ C|wn|q−1 + ∥un∥2−q q ∥un∥2−q X · g(x,wn∥un∥X , λn) ∥un∥q−1 X ≤ C|wn|q−1 + C2−qC|wn|p−1∥un∥p−1 X ∥un∥q−1 X = C1|wn|q−1 + C2∥un∥p−q X |wn|p−1 where C1 and C2 are constants that are independent of ∥un∥X . Considering that p− q > 0, it can be inferred that Theorem 2.10 remains valid when h is substituted with ∥u∥2−q q g(x, u, λ). This constitutes the subsequent theorem. Theorem 4.3. Let {(λn, un)}∞n=1 be solutions of (4.3). Then the following three stateents are equivalent, as n→ ∞: (i) ∥un∥W 1,2 0 (Ω) → 0, (ii) ∥un∥L∞(Ω) → 0, (iii) ∥un∥C1,β(Ω) → 0. Similar to Theorem 1.1, we have Theorem 4.1. Moreover, similar to Theorem 1.2, we can draw conclusions regarding bifurcation from infinity. Theorem 4.4. Let q ∈ [1, 2] and g satisfy (H8). Then the pair (λ1,∞) is a bifurcation point of (4.3). Moreover, there is a component D of the set of nontrivial solutions of (4.3) in R×X which meets (λ1,∞). If Λ ⊂ R is an interval such that Λ ∩ r(L) = {λ1} where r(L) is the set of real eigenvalue values of (1.2) and M is a neighborhood of (λ1,∞) whose projection on R lies in Λ and whose projection on X is bounded away from 0, then either (i) D − M is bounded in R×X in which case D − M meets R = {(λ, 0) | λ ∈ R}, or (ii) D − M is unbounded. If (ii) occurs and D − M has a bounded projection on R, then D − M meets (λ̂,∞) where λ1 ̸= λ̂ ∈ r(L). Similar to Corollary 3.2, we have the following statement. EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 15 Corollary 4.5. Let C be a component of the set of nontrivial solutions of (4.3) in R × X in Theorem 4.1 and D is a component of the set of nontrivial solutions of (4.3) in R×X in Theorem 4.4. (i) If (µn, un) ∈ C and is near (λ1, 0), then un = snφ1 +wn, where wn = o(|sn|) as |sn| → 0 and ∫ Ω wnφ1dx = 0. (ii) If (µn, un) ∈ D and is near (λ1,∞), then un = snφ1+wn, where wn = o(|sn|) as |sn| → ∞ and ∫ Ω wnφ1dx = 0. We will now proceed with the examination of (4.1). Let E = {u ∈ C1(Ω) : u = 0 on ∂Ω} with the usual norm ∥u∥C1 = max Ω |u|+max Ω |∇u|. Set P+ = { u ∈ E : u > 0 in Ω and ∂u ∂ω < 0 on ∂Ω } where ω is the outward pointing normal vector to ∂Ω. Lemma 4.6. Assume (H4) and (H5) hold. Then (λ1/f0, 0) is a bifurcation point of (4.1) and the associated bifurcation branch C ⊂ R×E whose closure contains (λ1/f0, 0) is either unbounded or contains a pair (λ/f0, 0) where λ is an eigenvalue of (1.2) and λ ̸= λ1. The above lemma is an application of Theorem 4.1. Lemma 4.7. Assume (H4) and (H5) hold. Then C ⊆ ((R × P+) ∪ (λ1/f0, 0)) and the last alternative in Lemma 4.6 is impossible. Proof. By the strong maximum principle [8], any nontrivial solution (λ, u) belongs to R×P+. So we have C ⊆ ((R×P+)∪(R×{0})). Suppose on the contrary that there exists (λn, un) → (λ/f0, 0) with (λn, un) ∈ C , un ̸= 0 and λ ̸= λ1. Let vn = un/∥un∥∞, then (λn, vn) satisfies∫ Ω ∇vn∇ϕ = λnf0 (∫ Ω |vn|q ) 2 q−1 ∫ Ω |vn|q−2vnϕ+ λn ∥un∥2−q q ∥un∥2−q ∞ ∫ Ω ξ(un) ∥un∥q−1 ∞ ϕ for all ϕ ∈ X. We obtain that there exists a subsequence vm → v as m→ +∞. Now v verifies the equation ∫ Ω ∇v∇ϕdx = λ̄ (∫ Ω |v|qdx ) 2 q−1 ∫ Ω |v|q−2vϕ dx. Hence v must change sign, and this is a contradiction. Furthermore, it follows that C ⊆ (R × P+) ∪ (λ1/f0, 0) and C is unbounded in R× E. □ Using (H6), we define η : R+ → R+ as f(s) = f∞|s|q−2s+ η(s) with lim s→+∞ η(s) |s|q−2s = 0. Then (4.1) is transformed into −∆u = λf∞ (∫ Ω |u|qdx ) 2 q−1 |u|q−2u+ λ (∫ Ω |u|qdx ) 2 q−1 η(u) in Ω, u = 0 on ∂Ω, (4.4) as a bifurcation problem from infinity. Similar to Lemma 4.6, 4.7, we have the following statement. Lemma 4.8. Assume (H4) and (H6) hold. Then (λ1/f∞,∞) is a bifurcation point of (4.1). Moreover, there exists a continuum D ⊂ ((R × P+) ∪ (λ1/f∞,∞)) of solutions of problem (4.1) meeting (λ1/f∞,∞) and satisfying at least one of the alternatives of Theorem 4.4. 16 Q. LIU, L. ZHAO EJDE-2025/105 Proof of Theorem 1.3. Firstly, we define f̃(s) = { f(s) if 0 ⩽ s ⩽ τ, 0 otherwise and consider the problem −∆u = λ (∫ Ω |u|qdx ) 2 q−1 f̃(u) in Ω, u = 0 on ∂Ω. (4.5) Utilizing Lemma 4.7, there exists a continuum C of nontrivial solutions of problem 4.5 ema- nating from (λ1/f0, 0) such that C ⊂ ((R× P+) ∪ (λ1/f0, 0)), meets ∞ in R×E. We claim that u ⩽ τ for any (λ, u) ∈ C . Suppose, by contradiction, that there exists x ∈ Ω so that u(x) > τ . Since u ∈ C1(Ω), we can find Ω1 ⊂ Ω so that u(x) > τ in Ω1 and u(x) = τ on ∂Ω1. This leads to the equation −∆u = 0 in Ω1, u = τ on ∂Ω1. This leads to a contradiction, as it follows that u(x) = τ in Ω1 by maximum principle. This substantiates the assertion and consequently indicates that u is also a solution of (4.1) for any (λ, u) ∈ C . Subsequently, we will demonstrate that the projection of C on R is unbounded. It is adequate to demonstrate that the set {(λ, u) ∈ C : λ ∈ (0, d]} is bounded for any fixed d ∈ (0,+∞). Arguing by contradiction, if there exists (λn, un) ∈ C , such that λn → λ′ ⩽ d, un → +∞ as n→ +∞. Let wn = un/∥un∥C1 . Then we have that wn = Q ( λn ∥un∥2−q q ∥un∥2−q C1 f̃(un) ∥un∥q−1 C1 ) . where Q = (−∆)−1. Clearly, we have that f̃(un) ⩽ max [0,τ ] |f(s)|. It means that λn ∥un∥2−q q ∥un∥2−q C1 f̃(un) ∥un∥q−1 C1 → 0 as n→ +∞. By the compactness of Q, we obtain that for some subsequence wn → 0 as n → +∞. This result stands in contradiction to the assertion that ∥wn∥C1 = 1. This, in conjunction with the observation that C joins (λ1/f0, 0) to infinity, indicates that (λ1/f0,+∞) ⊆ Proj(C ) where Proj(C ) denotes the projection of C on R. Applying Lemma 4.8, there exists a continuum D ⊂ (R × P+) ∪ (λ1/f∞,∞) of solutions of problem (4.1) meeting (λ1/f∞,∞) and satisfying at least one of the alternatives of Theorem 4.4. In addition, it is relatively straightforward to confirm that (λ1/f∞,∞) is the unique bifurcation point of positive solutions of (4.1) from ∞. We will demonstrate that these two components, C and D , are disjoint. Let F = {u ∈ C(Ω) : u = 0 on ∂Ω} with the usual norm ∥u∥∞ = max Ω |u|. It is sufficient to show that C and D are disjoint in R × F . We first claim that D is unbounded in the direction of F . And this only requires proving that (λ1/f∞, 0) is a blow-up point of D in R × F . Otherwise, there exists M > 0 such that ∥un∥∞ ⩽ M for any (λn, un) ∈ D with λn → λ1/f∞ as n → +∞. Applying [8, Theorem 8.33 of], we obtain that ∥un∥C1 ⩽ M ′ for some positive constant M ′, which contradicts the fact of D meeting (λ1/f∞,∞). Let us assume, for the sake of contradiction, that C ∩ D ̸= ∅ in R× F . Since D is unbounded in the direction of F , EJDE-2025/105 BIFURCATION FOR SEMILINEAR EIGENVALUE PROBLEMS 17 there exists (λ∗, u∗) ∈ (C ∩ D) such that maxΩ u ∗ = τ . Due to (H4), there exists 0 < κ < +∞ such that f(s) ⩽ κ(τ − s) for any s ∈ [0, τ ]. So we have −∆(τ − u∗) + λ∗κ∥u∗∥2−q q (τ − u∗) ≥ 0 in Ω, τ − u∗ > 0 on ∂Ω. The strong maximum principle of [8] implies that τ > u in Ω. This statement presents a paradox. Thus D − M is unbounded, where M is a neighborhood of (λ1/f∞,∞) whose projection on R contains λ1/f∞ and whose projection on E is bounded away from 0. We claim that D − M has an unbounded projection on R. This only needs to examine that the case of D − M meeting (λj/f∞,∞) for some j > 1 is not feasible, where λj denotes the eigenvalue of jth of eigenvalue problem (1.2). If not, we assume that D − M meets (λj/f∞,∞) for some j > 1. So there exists a neighborhood Ñ ⊂ M̃ of (λj/f∞,∞) such that u must change sign for any (λ, u) ∈ ((D − M ) ∩ (Ñ \ (λj/f∞,∞))), where M̃ is a neighborhood of (λj/f∞,∞) which satisfies the assumptions of Lemma 4.8. This contradicts that D ⊂ ( (R×P+)∪(λ1/f∞,+∞) ) . The anticipated conclusions are now evident, see Figure 1. □ 5. Appendix: Browder-Petryshyn degree We need some theories on the Browder-Petryshyn degree. We list some of its definitions and properties. For detailed information, we refer readers to [11, 12]. Now let X be a Banach space and D is a subset of X. We consider an operator A, in general nonlinear, defined on a subset of X, with values in X ′. Definition 5.1 (condition (S)+ or condition α(D)). Operator A belongs to the class (S)+ if for any sequence un ∈ D, un ⇀ u0 and limn→∞⟨Aun, un − u0⟩ ⩽ 0 imply un → u0. Definition 5.2. The operator A is said to be demicontinuous on D, if for any sequence un ∈ D strongly converging to u0 ∈ D, we have the equality lim n→∞ ⟨Aun, v⟩ = ⟨Au0, v⟩ for all v ∈ X. Definition 5.3. For F ⊂ D we denote by A(D,F ) the set of all bounded demicontinuous mappings A : D 7→ X ′ satisfying condition condition α(F ). When F = D, we write A(D) instead of A(D,D). We define Deg(A,D, 0) –the degree of a mapping A on the set D with respect to the origin of the space X ′– under the conditions: (a) A ∈ A(D), (b) Au ̸= 0 for any element u ∈ ∂D. Let {vi}, i = 1, 2, . . ., be any complete system of the space X and suppose that for every n the elements v1, . . . , vn are linear independent. Denote by Fn the linear hull of the elements v1, . . . , vn. Define for every n = 1, 2, . . . the finite-dimensional approximation An of the mapping A in the following way: Anu = n∑ i=1 ⟨Au, vi⟩vi for u ∈ Dn, Dn = D ∩ Fn. (5.1) Lemma 5.4. Let A be an operator satisfying conditions (a), (b). Then there exists N such that for n ⩾ N the following assertions hold: (1) the equation Anu = 0 has no solutions belonging to ∂Dn; (2) Leray-Schauder degree deg(An, Dn, 0) of the mapping An on the set Dn with respect to 0 ∈ Fn is defined and independent of n. By Theorem 5.4, limn→∞ deg(An, Dn, 0) exists and we denote it by D{vi}. Lemma 5.5. Suppose that the conditions (a), (b) are satisfied. Then the limit D{vi} = lim n→∞ deg(An, Dn, 0) does not depend on the the choice of the sequence {vi}. 18 Q. LIU, L. ZHAO EJDE-2025/105 The 5.4 and 5.5 justify the following definition. Definition 5.6 (Browder-Petryshyn degree). For an operator A satisfying conditions (a) and (b), by its degree on the set D with respect to the point 0 ∈ X ′ we mean the number lim n→∞ deg(An, Dn, 0), where An, Dn are determined in accordance with (5.1). This degree is denoted by Deg(A,D, 0). The degree of a mapping, introduced above, possesses all the natural properties of the degree of finite-dimensional mappings. Definition 5.7 (index of isolated zero point). The number lim r→0 Deg(A,Br(u0), 0) is called the index of the mapping A at the isolated zero point u0 and is denoted by Ind(A, u0). Lemma 5.8. Suppose that a mapping A of class A(D) has only isolated zero points in D and Au ̸= 0 for u ∈ ∂D. Then there exists only a finite number of zero points and the equality Deg(A,D, 0) = I∑ i=1 Ind(A, ui), holds, where ui, i = 1, . . . , I, are all zero points of the mapping A in D. Acknowledgments. The authors wish to express their gratitude to the anonymous referee for reading the paper carefully and making several corrections and remarks. References [1] A. Ambrosetti, D. Arcoya; An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser, Boston, Basel, Berlin, 2011. [2] A. Ambrosetti, A. Malchiodi; Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007. [3] T. Carroll, J. Ratzkin; Interpolating between torsional rigidity and principal frequency, J. Math. Anal. Appl. 379(2011), 818-826. [4] G. W. Dai; Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations, Topol. Methods Nonlinear Anal. 48(1)(2016), 213-233. [5] P. Drábek; Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics Series, Vol.264. Longman Scientific & Technical, 1992. [6] S. Fuč́ık, J. Nečas, J. Souček, V. Souček; Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathe- matics Vol. 346. Springer-Verlag, New York, Berlin, Heidelberg, 1973. [7] P. Girg, P. Takáč; Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems, Ann. Henri Poincaré. 9 (2008), 275-327. [8] D. Gilbarg, N. S. Trudinger; Elliptic partial differential equations of second order, Springer, Berlin, 2001. [9] P. H. Rabinowitz; Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7(1971), 487-513. [10] P. H. Rabinowitz; On bifurcation from infinity, J. Differential Equations. 14 (1973), 462-475. [11] I. V. Skrypnik; Nonlinear Elliptic Boundary Value Problems (in Russian), Naukovaja Dumka, Kyiev, 1973. English Translation: in Teubner-Texte zur Mathematik, Vol. 91. Teubner-Verlag, Leipzig, 1986. [12] E. Zeidler; Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, Berlin, Heidelberg, New York, 1990. Qingbo Liu School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address: liuqingbo@mail.dlut.edu.cn Lan Zhao (corresponding author) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address: zhaolan@mail.dlut.edu.cn 1. Introduction 2. Preliminaries 2.1. Properties of the first eigenvalue 1 2.2. Equivalence of norms near (1,0) and (1,) 3. Global bifurcation 4. Positive solutions 5. Appendix: Browder-Petryshyn degree Acknowledgments References