Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 239–253. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu CONNECTED COMPONENTS OF POSITIVE SOLUTIONS OF BIHARMONIC EQUATIONS WITH THE CLAMPED PLATE CONDITIONS IN TWO DIMENSIONS RUYUN MA, ZHONGZI ZHAO, DONGLIANG YAN In memory of Professor Alan C. Lazer Abstract. This article concerns the clamped plate equation ∆2u = λa(x)f(u), in Ω, u = ∂u ∂ν = 0 on ∂Ω, where Ω is a bounded domain in R2 of class C4,α, a ∈ C(Ω̄, (0,∞)), f : [0,∞) → [0,∞) is a locally Hölder continuous function with exponent α, and λ is a positive parameter. We show the existence of S-shaped connected com- ponent of positive solutions under suitable conditions on the nonlinearity. Our approach is based on bifurcation techniques. 1. Introduction Let Ω denote a bounded domain in R2 of class C4,α. We consider the clamped plate problem ∆2u = λf̃(x, u) in Ω, (1.1) u = ∂u ∂ν = 0 on ∂Ω, (1.2) where ∂/∂ν is the outward normal derivative, α ∈ (0, 1], f̃ : Ω̄× [0,∞)→ [0,∞) is a locally Hölder continuous function with exponent α. (1.1), (1.2) forms a model for the clamped plate where f̃ is the load and u the deviation of the plate Ω. Boggio [2, 3] and Hadamard [16, 17] extensively studied this model when λf̃(x, u) = e(x) and f̃(x, u) = u, respectively. Dalmasso [7] used the Schauder fixed point theorem to study the existence of positive solutions of nonlinear boundary-value problem of elliptic equation of order 2m under the assumptions (1) for x ∈ Ω, f̃(x, s) is nondecreasing in s; (2) lims→0 minx∈Ω̄ f̃(x, s)/s =∞, lims→∞maxx∈Ω̄ f̃(x,s) s = 0, 2010 Mathematics Subject Classification. 35J40, 35G30, 35B32, 35P30. Key words and phrases. Biharmonic operator; positive solutions; eigenvalue; bifurcation. ©2021 This work is licensed under a CC BY 4.0 license. Published November 3, 2021. 239 240 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 and considered the following domains: the unit ball B = {x ∈ RN : ‖x‖ < 1}, N ≥ 1, and a bounded domain of class C2m,α close in C2m,α-sense to a ball. Mâagli, Toumi, and Zribi [20] also used the Schauder fixed point theorem to show the existence of positive continuous solution (in the sense of distributions), when Ω is the unit ball B in RN and N ≥ 2, and the nonlinearity f̃ satisfies appropriate conditions related to a Kato class of functions Km,N . At most two radial positive solutions were obtained in above mentioned papers. The aim of this article is to study the global structure of positive solutions for problem (1.1), (1.2) on Ω ⊂ R2 when f̃(x, s) = a(x)f(s), x ∈ Ω̄, s ∈ [0,∞), and to show that the positive solutions set contains an S-shaped connected com- ponent under suitable conditions; consequently, (1.1), (1.2) possesses at least three positive solutions for λ belonging to certain open interval. We work on Ω ⊂ R2 for the following two reasons: (1) we need to assume that Ω is a bounded domain of class C4,α(Ω̄) which is ε0-close in C4,α-sense to B ⊂ R2 for some ε0 > 0 (see Grunau and Sweers [13, 14] for the detail); (2) Harnack inequalities are very important in study of the shape of connected components of positive solutions of second order elliptic problems, see Sim and Tanaka [23]. However, no general Harnack inequalities are available for the poly- harmonic problems, see Gazzola, Grunau, and Sweers [11, P.146]. Caristi and Mitidieri [6, Theorem 3.6] proved a Harnack type inequalities for linear biharmonic equations containing a Kato potential when N > 4, which cannot be used to treat the biharmonic problem on Ω ⊂ R2. To establish a Harnack inequality for bihar- monic problems on Ω ⊂ R2, we need (4.13) below. Notice that (4.13) need the restriction N = m = 2. For earlier results on the existence and multiplicity of solutions to the mathe- matical models of nonlinearly supported bending beams see the well-known survey paper of Lazer and Mckenna [18]. 2. Preliminaries Let Y be the Banach space C(Ω̄) equipped with the supremum norm ‖ · ‖C(Ω̄). 2.1. Principal eigenvalue. The biharmonic eigenvalue problem with Dirichlet boundary conditions has the form ∆2ϕ = λϕ in Ω, ϕ = ∂ϕ ∂ν = 0 on ∂Ω. (2.1) The famous conjecture for this problem was as follows; by now it has numerous counterexamples. Conjecture (Szegö, 1950) If Ω is a ‘nice’ domain (convex), then the first eigen- function for (2.1) is of fixed sign. This conjecture was proved to be wrong, see Duffin and others [8, 10, 19, 4, 22]. Coffman [4] proved that the first eigenfunction on a square changes sign. For the domains Aε = {(x, y) ∈ R2 : ε2 < x2 + y2 < 1} with 0 < ε < 1. EJDE-2021/SI/01 BIHARMONIC EQUATIONS 241 Coffman, Duffin and Shaffer [5] proved the fundamental mode of vibration of a clamped annular plate Aε is not of one-sign. We first recall the definition of closeness of domain introduced by Grunau and Sweers [13]. Definition 2.1. Let ε > 0, α ∈ (0, 1], Ω is called ε-closed in Ck,α-sense to Ω∗, if there exists a Ck,α mapping g : Ω̄∗ → Ω̄ such that g(Ω̄∗) = Ω̄ and ‖g − Id‖Ck,α(Ω̄∗) ≤ ε. Using Dalmasso [7, Lemma 3.1(2)] and Dalmasso [7, Theorem 2.2 (ii)], we may deduce the following result. Lemma 2.2. Let Ω ⊂ R2 and Ω is a bounded domain of class C4,α. Then there exists ε0 > 0 such that if Ω is ε-close in C4,α sense to B for all 0 < ε ≤ ε0, then (1) the problem ∆2u = e in Ω, u = ∂u ∂ν = 0 on ∂Ω with some e ∈ C0,α(Ω̄) has unique solution u ∈ C4,α(Ω̄). (2) If e ≥ 0 and e 6≡ 0, then ∂2u ∂ν2 > 0 for x ∈ ∂Ω. In the following, we consider the eigenvalue problem ∆2u = λa(x)u, in Ω, u = ∂u ∂ν = 0 on ∂Ω, (2.2) where a ∈ C(Ω̄, (0,∞)). The first eigenvalue of (2.2) is defined as λ1(a(·)) = min u∈H2 0 (Ω)\{0} ‖∆u‖2 H2 0 ‖a1/2u‖2L2 , where H2 0 (Ω) is the closure of C∞c (Ω) with respect to the normal ‖ · ‖W 2,2 , and C∞c (Ω) is the space of C∞(Ω)-functions having compact support in Ω. Applying Lemma 2.2 and the standard Krein-Rutman type argument, we may obtain the following result. Lemma 2.3. Let ε0 be the constant as given in Lemma 2.2. If Ω ⊂ R2 and Ω is a bounded domain of class C4,α(Ω̄) which is ε0-close in C4,α-sense to B, then (1) the first eigenvalue λ1(a(·)) of (2.2) is simple; (2) the corresponding eigenfunction ψ is of one sign; (3) ∂2ψ ∂ν2 > 0, x ∈ ∂Ω. 2.2. Shape of positive solutions. We will make the following assumptions: (H0) f : [0,∞) → [0,∞) is a Hölder continuous function with exponent α, and f(s) > 0 for s > 0; (H1) a ∈ C(Ω̄, (0,∞)); (H2) there exist β > 0, f0 > 0 and f1 > 0 such that lim s→0+ f(s)− f0s s1+β = −f1; 242 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 (H3) f∞ := lim s→∞ f(s) s = 0. Remark 2.4. It is easy to show that if (H2) holds, then lim s→0+ f(s) s = f0. Moreover, if (H3) holds, then there exists s̃ > 0, f∗ > 0 and γ∗ > 0 such that f(s) ≤ f∗s, ∀s ≥ 0; f(s) ≥ γ∗s, ∀s ∈ [0, s̃]. (2.3) Lemma 2.5. Let (H0)–(H2) hold. Let s0 ∈ (0,∞) be a constant and let (λ, u) be the nonnegative solution of ∆2u = λa(x)f(u) x ∈ Ω, u = ∂u ∂ν = 0 x ∈ ∂Ω (2.4) with max{u(x) : x ∈ Ω̄} = u(x0) = s0. Then λ ∈ (0,M1] for some positive constant M1 > 0, which is independent of u and λ. Proof. Assume on the contrary that there exists a sequence {(µn, un)} of positive solutions of (2.4) with ‖un‖C(Ω̄) = s0, µn →∞ as n→∞. (2.5) Let yn := un/‖un‖C(Ω̄). Then ∆2yn = µna(x) f(un(x)) un(x) yn x ∈ Ω, yn = ∂yn ∂ν = 0 x ∈ ∂Ω. (2.6) Since (H0) and (H2) imply that f(s)/s ≥ ρ0 for s ∈ (0, s0] for some ρ0 > 0, we let ψ : ψ(x) > 0 in Ω, be the eigenfunction corresponding λ1(a(·)), i.e. ∆2ψ = λ1(a(·))a(x)ψ, in Ω, ψ = ∂ψ ∂ν = 0 on ∂Ω. (2.7) Multiplying the equation in (2.6) by ψ and multiplying the equation in (2.7) by yn, integrating over Ω by parts and using that∫ Ω ψ ∆2yndx = ∫ Ω ∆yn∆ψ dx, (2.8) we deduce from µn → ∞ that yn must change its sign in Ω if n is large enough. However, this is a contradiction. � Lemma 2.6. Let (H0)–(H2) hold. Let s0 ∈ (0,∞) be a constant and let Λ :=[ 0,max{M1, λ1(a(·))/f0 + 1} ] be a compact interval. Let (λ, u) be the nonnegative solution of ∆2u = λa(x)f(u) x ∈ Ω, (2.9) u = ∂u ∂ν = 0 x ∈ ∂Ω, (2.10) EJDE-2021/SI/01 BIHARMONIC EQUATIONS 243 with λ ∈ Λ and max{u(x) : x ∈ Ω̄} = u(x0) = s0. Then x0 ∈ Ωδ := {x ∈ Ω : d(x, ∂Ω) ≥ δ} (2.11) for some positive constant δ = δ(s0), which is independent of λ ∈ Λ. Proof. Assume on the contrary that there exists a sequence {(µk, yk)} of nonnega- tive solutions of (2.9), (2.10) with µk ∈ Λ, ‖yk‖C(Ω̄) = s0 and d(x0,k, ∂Ω)→ 0 as k →∞, where yk(x0,k) = max{yk(x) : x ∈ Ω̄}. Since {µka(·)f(yk(·))} is uniformly bounded in C(Ω̄), it follows that ‖µka(·)f(yk(·))‖Lp(Ω) ≤M2 (2.12) for some constant M2 > 0. By Agmon-Douglis-Nirenberg estimates in [1], for any p > 1, ‖uk‖W 4,p(Ω) ≤ Cp‖µka(·)f(yk(·))‖Lp(Ω) ≤ CpM2, (2.13) where Cp is a positive constant. By the embedding theorem [11, Theorem 2.6], W 4,p(Ω) ↪→ C3,α(Ω̄) for all p > 2 4−3 = 2 and α ∈ (0, 1− 2 p ] ∩ (0, 1). Thus ‖uk‖C3,α(Ω̄) ≤M3 (2.14) for some constant M3 > 0. Since C3,α(Ω̄) ↪→↪→ C(Ω̄) is a compact embedding, it follows that after taking a subsequence if necessary, yk converges to ŷ in C(Ω̄). Moreover, ‖ŷ‖C(Ω̄) = s0. (2.15) Since Ω̄ ⊂ R2 is bounded and closed, we may assume that x0,k → x∗, and conse- quently, ŷ(x∗) = s0. On the other hand, x∗ ∈ ∂Ω, which together with the fact yn(x) = 0 on ∂Ω imply ŷ(x∗) = 0. However, this contradicts (2.15). � 2.3. Global solutions branches for positive mappings. Suppose that E is a real Banach space with norm ‖ · ‖. Let K be a cone in E. A nonlinear mapping A : [0,∞)×K → E is said to be positive if A([0,∞)×K) ⊆ K. It is said to be K- completely continuous if A is continuous and maps bounded subsets of [0,∞)×K to precompact subset of E. If L is a continuous linear operator on E, denote r(L) the spectral radius of L. Define cK(L) = {λ ∈ [0,∞) : there exists x ∈ K with ‖x‖ = 1 and x = λLx}. The following Lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [9, Theorem 2] . Lemma 2.7. Assume that (i) K has nonempty interior and E = K −K; (ii) A : [0,∞) ×K → E is K-completely continuous and positive, A(λ, 0) = 0 for λ ∈ R, A(0, u) = 0 for u ∈ K and A(λ, u) = λLu+ F (λ, u), where L : E → E is a strongly positive linear compact operator on E with r(L) > 0, F : [0,∞) × K → E satisfies ‖F (λ, u)‖ = ◦(‖u‖) as ‖u‖ → 0 locally uniformly in λ. 244 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 Then there exists an unbounded connected subset C of DK(A) = {(λ, u) ∈ [0,∞)×K : u = A(λ, u), u 6= 0} ∪ {(r(L)−1, 0)} such that (r(L)−1, 0) ∈ C. 3. Main results Let s̃ be a positive constant. In the rest of this paper we will take δ to be the constant in Lemma 2.6 with Λ = [0,max{M1, λ1(a(·))/f0 + 1}]. To study the multiplicity of positive solutions of (2.9),(2.10), we need the following assumption (H4) min s̃ C≤s≤s̃ f(s) s > Cf0 λ1(a(·)) minΩδ/2 G2,2,Ω(x, y)a0|Bδ/2| , (3.1) where a0 = minΩ̄ a(·), |Bδ/2| = measBδ/2, Ωr := {x ∈ Ω : d(x, ∂Ω) > r}, Br := {x ∈ B : d(x, ∂B) > r}, and C is the constant satisfying 1 C (d(x))2G2,2,B(0, y) ≤ G2,2,B(x, y) ≤ CG2,2,B(0, y) x, y ∈ B, (3.2) where d(x) = d(x, ∂Ω), G2,2,B is the Green function of ∆2 for the Dirichlet problem in B, see Mâagli, Toumi and Zribi [20, P.3] for the details. Using a similar idea to show the existence of three positive solutions of one- dimensional p-Laplacian problem and arguing the shape of bifurcation as in Sim and Tanaka [23], we have the following results for ∆2u = λa(x)f(u) in Ω, (3.3) u = ∂u ∂ν = 0 on ∂Ω. (3.4) Theorem 3.1. Let ε0 be the constant in Lemma 2.2. Let Ω ⊂ R2 is a bounded domain of class C4,α(Ω̄) which is ε0-close in C4,α-sense to B. Let (H0)–(H4) hold. Then there exist λ∗ ∈ (0, λ1(a(·))/f0) and λ∗ ∈ (λ1(a(·))/f0,∞) such that (i) (3.3), (3.4) has at least one positive solution if λ = λ∗; (ii) (3.3),(3.4) has at least two positive solutions if λ∗ < λ ≤ λ1(a(·))/f0; (iii) (3.3), (3.4) has at least three positive solutions if λ1(a(·))/f0 < λ < λ∗; (iv) (3.3), (3.4) has at least two positive solutions if λ = λ∗; (v) (3.3), (3.4) has at least one positive solution if λ > λ∗. See illustrations in Figure 1. Remark 3.2. From Grunau and Sweers [14, 15], the Green function in (3.2) is G2,2,B(x, y) = k2,2|x− y|2 ∫ ∣∣|x|y− x |x| ∣∣/|x−y| 1 (v2 − 1)v−1dv, x, y ∈ B, (3.5) and satisfies G2,2,B(x, y) ∼ d(x)d(y) min { 1, d(x)d(y) |x− y|2 } , (3.6) where k2,2 is a known constant. By combining (3.5), (3.6) and doing numerical calculation, the exact value of C in (H4) can be obtained, denoted as C�. EJDE-2021/SI/01 BIHARMONIC EQUATIONS 245 - 6‖u‖∞ λ∗ λ1((a(t)) f0 λ∗ λ Figure 1. Connected component of the solution set of (3.3), (3.4) Remark 3.3. For the general case Ω 6= B, we may transform (3.3), (3.4) into a new problem in B using the holomorphic mapping from Ω to B, see Grunau and Sweers [15]. By (3.5) and some simple computations, we may obtain a constant C∗ > 0 such that the Green function G2,2,Ω(x, y) of (3.3), (3.4) and G2,2,B(x, y) satisfy 1 C∗ G2,2,B(x, y) ≤ G2,2,Ω(x, y) ≤ C∗G2,2,B(x, y). Remark 3.4. We may provide an example to illustrate the application of Theorem 3.1 in the case Ω = B. Take K = max {1 2 , C� λ1(1) G̃δ/2 |Bδ/2| } + 1 and G̃δ/2 := minBδ/2 G2,2,B(x, y). Let us consider the boundary value problem ∆2u = f̂(u), in B, u = ∂u ∂ν = 0 on ∂B, (3.7) with f̂(s) =  s− s2, if s ∈ [0, 1/2), (2K − 1 2 )s−K + 1 2 , if s ∈ [1/2, 1), Ks2, if s ∈ [1, C�], K(C�)3/2 √ s, if s ∈ (C�,∞). Obviously, f̂ is a continuous, non-decreasing function with f(0) ≥ 0, from [11, Theorem 7.1] the solution u of (3.7) is radially symmetric. So, we may take δ = 1/4. Obviously, f̂ satisfies (H2) and (H3) with β = 1, f1 = 1, f0 = 1; (H4) with s̃ = C� is satisfied since min s̃ C�≤s≤s̃ f(s) s = min 1≤s≤C� Ks > K > C� λ1(1)G̃1/8 |B1/8| . 246 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 Thus, we are in the position to use Theorem 3.1. 4. Bounds of solutions 4.1. A priori estimation. Let X = { u ∈ C2,α(Ω̄) : u satisfies (3.4), and there exists γ ∈ (0,∞) such that − γψ(x) ≤ u(x) ≤ γψ(x), x ∈ Ω } . (4.1) Then X is a Banach space under the norm ‖u‖X := inf{γ : −γψ(x) ≤ u(x) ≤ γψ(x) for x ∈ Ω}. Let P := {u ∈ X : u(x) ≥ 0, x ∈ Ω}. (4.2) Then P is normal, has a nonempty interior, and X = P − P . Lemma 4.1. Let Ω be as in Theorem 3.1. Let (H0)–(H3) hold. Let J := [a1, b1] ⊂ [0,∞). Assume that {(µn, yn)} be a sequence of solutions of (3.3),(3.4) with µn ∈ J, ‖yn‖C(Ω̄) ≤M (4.3) for some constant M , independent of n. Then yn ∈ C4(Ω̄)∩X and {yn} is bounded in X. Proof. It follows from (2.3), ∆2yn = µna(x)f(yn) in Ω, yn = ∂yn ∂ν = 0 on ∂Ω, and Grunau and Sweers [14, P.620], that for any p > 1, ‖yn‖W 4,p 0 (Ω) ≤M4 for some positive constant M4, independent of n. Thus, the Sobolev imbedding theorem [12, Corollary 7.1] guarantees that ‖yn‖C3(Ω̄) ≤M5, and consequently, ‖yn‖C0,α(Ω̄) ≤ M6 for some positive constant M6, independent of n. Thus ‖µnaf(yn)‖C0,α(Ω̄) ≤M7 for some positive constant M7, independent of n. Combining this with (3.3), (3.4) and using [7, Lemma 3.1], it follows that ‖yn‖C4,α(Ω̄) ≤M8 for some positive constant M8, independent of n. Therefore, |yn(x)| ≤ C8ψ(x) x ∈ Ω for some positive constant C8, independent of n. Therefore, ‖yn‖X ≤M9 for some positive constant M9, independent of n. � EJDE-2021/SI/01 BIHARMONIC EQUATIONS 247 Let h : B → Ω be a bijection such that h(x1 + ix2) = h1(x1, x2) + ih2(x1, x2) is a holomorphic mapping. Then ∆(u ◦ h) = 1 2 |∇h| 2(∆u) ◦ h. We write g(x) = 2|(∇h)(x)|−2. (4.4) If ∂Ω is sufficiently smooth, then a Theorem of Kellogg-Warschawski (see [21]) implies that h is sufficiently smooth and that there exist ci > 0 such that c1 ≤ |(∇h)(x)|−2 ≤ c2. The problem (3.3), (3.4) can be transformed into (g(·)∆)2(u ◦ h) = (λa(·)f(u) ◦ h) in B, (4.5) (u ◦ h) = ∂(u ◦ h) ∂ν = 0 on ∂B, (4.6) which can also be written as( (−∆)2 +A ) (u ◦ h) = g−2 ( (λa(·)f(u)) ◦ h ) in B, (4.7) (u ◦ h) = ∂(u ◦ h) ∂ν = 0 on ∂B, (4.8) where for some A of the form A = ∑ |α|<4 aα(x)Dα, aα ∈ C(B̄). (4.9) And Ω is close to the disk B means that ‖h − Id‖C3(B̄) sufficiently small. For example this holds for an ellipse that is close to a circle, see Grunau and Sweers[13]. Lemma 4.2. Let Ω be as in Theorem 3.1 and N = 2. Let I ⊂ (0,∞) be a compact interval. Assume that (H0)–(H3) hold. Then there exists M10 > 0, such that for any positive solutions of (3.3), (3.4) with λ ∈ I, we have ‖u‖C(Ω̄) ≤M10. (4.10) Proof. Suppose on the contrary that there exists a sequence {(µn, un)} of positive solutions of (3.3), (3.4), such that µn ∈ I, ‖un‖C(Ω̄) →∞. (4.11) This together with the fact h : B → Ω is a bijection and ‖h−Id‖C3(B̄) is sufficiently small that ‖un ◦ h‖C(B̄) →∞. (4.12) By Mâagli, Toumi and Zribi [20, P.3], N = m = 2 implies 1 C (d(x))2G2,2,B(0, y) ≤ G2,2,B(x, y) ≤ CG2,2,B(0, y) x, y ∈ B, (4.13) 248 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 where d(x) := dist(x, ∂B) > 0 in B. From this and (4.11), (4.12), it follows that for x ∈ B, (un ◦ h)(x) = λ ∫ B G2,2,B(x, y)af((un ◦ h)(y))dy ≥ λ ∫ B 1 C (d(x))2G2,2,B(0, y)af((un ◦ h)(y))dy ≥ λ ∫ B 1 C (d(x))2 1 C G2,2,B(xu, y)af((un ◦ h)(y))dy = ( 1 C )2 (d(x))2 ∫ B λG2,2,B(xu, y)af((un ◦ h)(y))dy = 1 C2 (d(x))2‖un ◦ h‖C(B̄), (4.14) where (u ◦ h)(xu) = ‖u ◦ h‖C(Ω̄). Thus, for any σ > 0, lim n→∞ (un ◦ h)(x) =∞ uniformly for x ∈ Ωσ. (4.15) Let yn := un ◦ h ‖un ◦ h‖C(B̄) . Then by (4.11), (4.12) and standard compact argument, we deduce that after taking a subsequence if necessary, yn → y∗ for some y∗ with ‖y∗‖C(B̄) = 1. On the other hand, combining (4.11), (4.12), and using f∞ = 0, I ⊂ [0,∞), and (4.15), it follows that ‖y∗‖C(B̄) = 0. However, this is a contradiction. � Using a similar argument for (4.14), we obtain the following Harnack type in- equalities. Lemma 4.3. Let Ω ⊂ R2 be as in Theorem 3.1. Let β1 and β2 ∈ (0,∞) be two positive constants. Let V ∈ C(Ω̄) with β1 ≤ V (x) ≤ β2 x ∈ Ω. If u is a nonnegative weak solution of ∆2u = V (x)u x ∈ Ω, u = ∂u ∂ν = 0 x ∈ ∂Ω, then for any σ > 0, there exists C = C(β1, β2) such that we have sup Ω̄ u ≤ C inf Ωσ u, where C is independent of u and V ∈ {w ∈ Y : β1 ≤ w(x) ≤ β2 for x ∈ Ω}. 5. Rightward bifurcation Define L : D(L)→ Y by Lu := ∆2u, on the domain D(L) = {u ∈ C2,α(Ω̄) ∩ C4(Ω) : u satisfies (3.4)}. It is easy to check that L−1 : Y → Y is compact. EJDE-2021/SI/01 BIHARMONIC EQUATIONS 249 It follows from Dalmasso [7, Theorem 2.3] that if for any z ∈ Y with z ≥ 0 and z(x0) > 0 for some x0 ∈ Ω̄ with Lu− z = 0. (5.1) Then u ∈ intP . Let ζ, ξ ∈ C([0,∞)) be such that f(u) = f0u+ ζ(u), f(u) = f∞u+ ξ(u) with lim u→0 ζ(u) u = 0, lim u→∞ ξ(u) u = 0. Let ξ̃(r) = max{|ξ(u)| : 0 ≤ u ≤ r}. (5.2) Then ξ̃ is nondecreasing and lim r→∞ ξ̃(r) r = 0. (5.3) Let us consider Lu(x) = λf0a(x)u(x) + λa(x)ζ(u(x)), x ∈ Ω̄ (5.4) as a bifurcation problem from the trivial solution u ≡ 0. Combining this with Lemma 2.7, we can conclude that there exists an unbounded connected subset C of the set {(λ, u) ∈ (0,∞)× P : (λ, u) satisfies (5.4), u ∈ intP} ∪ {(λ1(a(·))/f0, 0)} such that (λ1(a(·))/f0, 0) ∈ C. By the method used by Sim and Tanaka to prove [23, Lemma 2.3], with obvious changes, we obtain the following result. Lemma 5.1. Let Ω be as in Theorem 3.1. Let (H0)–(H2) hold. Let {(ηj , uj)} be a sequence of positive solutions to (3.3), (3.4) which satisfies ‖uj‖C(Ω̄) → 0 and ηj → λ1(a(·))/f0. Let ψ be the eigenfunction corresponding to λ1(a(·)), which satisfies ‖ψ‖C(Ω̄) = 1. Then there exists a subsequence of {uj}, again denoted by {uj}, such that uj/‖uj‖C(Ω̄) converges uniformly to ψ on Ω̄. Lemma 5.2. Let Ω be as in Theorem 3.1. Let (H0)–(H2) hold. Let C be as in Lemma 2.7. Then there exists δ̂ > 0 such that (λ, u) ∈ C and |λ − λ1(a(·))/f0| + ‖u‖C(Ω̄) ≤ δ̂ imply λ > λ1(a(·))/f0. Proof. Assume on the contrary that there exists a sequence {(ηj , uj)} such that (ηj , uj) ∈ C, ηj → λ1(a(·))/f0, ‖uj‖C(Ω̄) → 0 and ηj ≤ λ1(a(·))/f0. By the standard argument, we may get that there exists a subsequence of {uj}, again denoted by {uj}, such that uj/‖uj‖C(Ω̄) converges uniformly to ψ on Ω̄, where ψ > 0 is the first eigenfunction of (2.2) which satisfies ‖ψ‖C(Ω̄) = 1. Multiplying (3.3) with (λ, u) = (ηj , uj) by uj and integrating it over Ω, we obtain ηj ∫ Ω a(x)f(uj(x))uj(x)dx = ∫ Ω (∆uj(x))2dx. Using the definition of λ1(a(·)), we obtain ηj ∫ Ω a(x)f(uj(x))uj(x)dx ≥ λ1(a(·)) ∫ Ω a(x)(uj(x))2dx. 250 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 It is easy to see that∫ Ω a(x) f(uj(x))− f0uj(x) |uj(x)|1+β ∣∣∣ uj(x) ‖uj‖C(Ω̄) ∣∣∣2+β dx ≥ λ1(a(·))− f0ηj ηj‖uj‖βC(Ω̄) ∫ Ω a(x) ∣∣∣ uj(x) ‖uj‖C(Ω̄) ∣∣∣2 dx. Lebesgue’s dominated convergence theorem and (H2) imply that∫ Ω a(x) f(uj(x))− f0 uj(x) |uj(x)|1+β ∣∣∣ uj(x) ‖uj‖C(Ω̄) ∣∣∣2+β dx→ −f1 ∫ Ω a(x)|ψ(x)|2+βdx < 0 and ∫ Ω a(x) ∣∣∣ uj(x) ‖uj‖C(Ω̄) ∣∣∣2 dx→ ∫ Ω a(x)|ψ(x)|2 dx > 0. This contradicts ηj ≤ λ1(a(·))/f0. � 6. Direction turn of bifurcation In this section, we show that there is a direction turn of the bifurcation under assumptions (H3) and (H4). Lemma 6.1. Let Ω be as in Theorem 3.1. Let (H0)–(H3) hold. Let u ∈ C4(Ω̄) be the positive solution of (3.3), (3.4) with u(x0) = ‖u‖C(Ω̄) = s0 for some s0 > 0, and λ ∈ [0,max{M1, λ1(a(·))/f0 + 1}]. Then 1 C ‖u‖C(B̄δ/2(x0)) ≤ u(x) ≤ ‖u‖C(B̄δ/2(x0)), x ∈ Bδ/2(x0) (6.1) where C is the constant in (3.2). Proof. Lemma 2.6 yields x0 ∈ Ωδ. Thus the desired results is an immediate conse- quence of (4.13). � Lemma 6.2. Let Ω be as in Theorem 3.1. Assume that (H0)–(H4) hold. Let u be a positive solution of (3.3),(3.4) with ‖u‖C(Ω̄) = s0. Then λ < λ1(a(·))/f0, or λ > λ1(a(·))/f0 + 1. Proof. Let u be a positive solution of (3.3), (3.4). Then from Lemma 6.1 we have 1 C s0 ≤ u(x) ≤ s0, x ∈ Bδ/2(x∗), where u(x∗) = ‖u‖C(Ω̄). Assume on the contrary that λ ≥ λ1(a(·))/f0. Then from Lemma 2.6 and (H4), it follows that s0 = u(x∗) = λ ∫ Ω G2,2,Ω(x∗, y)a(y)f(u(y))dy ≥ λ ∫ Ωδ/2 G2,2,Ω(x∗, y)a(y)f(u(y))dy ≥ λ ∫ Bδ/2(x∗) G2,2,Ω(x∗, y)a(y)f(u(y))dy EJDE-2021/SI/01 BIHARMONIC EQUATIONS 251 ≥ λ ∫ Bδ/2(x∗) G2,2,Ω(x∗, y)a(y) f(u(y)) u(y) (u(y))dy ≥ λ1(a(·)) f0 min Ωδ/2 G2,2,Ω(x, y)a0 measBδ/2 min s0 C ≤s≤s0 f(s) s s0 C > s0. This is a contradiction. Therefore, λ < λ1(a(·)) f0 . � 7. Second turn and proof of Theorem 3.1 In this section, we give a block for a parameter and a priori estimate and finally a proof of Theorem 3.1. Lemma 7.1. Let Ω be as in Theorem 3.1. Assume that (H0)—(H4) hold. Let (λ, u) be a positive solution of (3.3),(3.4). Then there exists C1 > 0 independent of u such that λf(‖u‖C(Ω̄)) < C1, where f(s) := min s C≤t≤s f(t)/t. (7.1) Proof. Let u(xu) = ‖u‖C(Ω̄). Then u(xu) = λ ∫ Ω G2,2,Ω(xu, y)a(y)f(u(y))dy ≥ λ ∫ Bδ(xu) G2,2,Ω(xu, y)a(y)f(u(y))dy ≥ λmin Ωδ/2 G2,2,Ω(x, y)|Bδ|a0f(‖u‖C(Ω̄)) 1 C ‖u‖C(Ω̄), which implies λf(‖u‖C(Ω̄)) < C1 for some C1 > 0. � Proof of Theorem 3.1. By Lemma 5.2, C is bifurcating from (λ1(a(·))/f0, 0) and goes rightward. We claim that there exists a sequence {(βj , uj)} ⊂ C satisfying βj → +∞, ‖uj‖C(Ω̄) →∞. (7.2) Assume on the contrary that there exists β∗ > 0, such that ‖u‖C(Ω̄) ≤M11 for all (λ, u) ∈ C with λ > β∗. (7.3) Then 0 ≤ ‖u‖C(Ω̄) ≤ M11 implies f(‖u‖C(Ω̄)) ≥ δ0 for some constant δ0 > 0, and consequently λf(‖u‖C(Ω̄))→∞ as λ→∞. (7.4) However, this contradicts Lemma 7.1. Therefore, (7.2) holds. Thus, there exists (β0, u0) ∈ C such that ‖u0‖C(Ω̄) = s0. Lemma 6.2 implies that β0 < λ1(a(·))/f0. By Lemmas 5.2, 6.2 and 4.3, C passes through some points (λ1(a(·))/f0, v1) and (λ1(a(·))/f0, v2) with ‖v1‖C(Ω̄) < s0 < ‖v2‖C(Ω̄). By Lemmas 5.2 and 6.2 and the fact C ∩ ({0} × P ) = {(0, 0)}, there exist λ̄ and λ which satisfy 0 < λ < λ1(a(·))/f0 < λ̄ and both (i) and (ii): (i) if λ ∈ (λ1(a(·))/f0, λ̄], then there exists u and v such that (λ, u), (λ, v) ∈ C and ‖u‖C(Ω̄) < ‖v‖C(Ω̄) < s0; 252 R. MA, Z. ZHAO, D. YAN EJDE/SI/01 (ii) if λ ∈ (λ, λ1(a(·))/f0], then there exists u and v such that (λ, u), (λ, v) ∈ C and ‖u‖C(Ω̄) < s0 < ‖v‖C(Ω̄). Define λ∗ = sup{λ̄ : λ̄ satisfies (i)} and λ∗ = inf{λ : λ satisfies (ii)}. Then by the standard argument, (3.3), (3.4) has a positive solution at λ = λ∗ and λ = λ∗, respectively. Since C passes through (λ1(a(·))/f0, v2) and (βj , uj), Lemma 6.2 and 2.7 imply that, for each λ > λ1(a(·))/f0, there exists w such that (λ,w) ∈ C and ‖w‖C(Ω̄) > s0. This completes the proof. � Acknowledgements. R. Ma was supported by NSFC (No.12061064). The au- thors are very grateful to the anonymous referees for their valuable suggestions. References [1] S. Agmon, A. Douglis, L. Nirenberg; Estimates near the boundary for solution of elliptic par- tial differential equations satisfying general boundary conditions I. Comm Pure Appl Math., 12(1959), 623-727. [2] T. Boggio; Sull’equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei, 10 (1901), 197-205. [3] T. Boggio; Sulle funzione di Green dórdine m, Rend. Circ. Mat. Palermo, 20(1905), 97-135. [4] C. V. Coffman; On the structure of solutions to ∆2u = λu which satisfy the clamped plate conditions on a right angle, SIAM J. Math. Anal., 13(1982), 746-757. [5] Ch. V. Coffman, R. J. Duffin, D. H. Shaffer; The fundamental mode of vibration of a clamped annular plate is not of one sign. In Constructive approaches to mathematical models (Proc. Conf. in honor of R.J. Duffin, Pittsburgh, Pa., 1978), pages 267-277. Academic Press, New York, London, 1979. [6] G. Caristi, E. Mitidieri; Harnack inequality and applications to solutions of biharmonic equa- tions. Partial differential equations and functional analysis, 1-26. Oper. Theory Adv. Appl. 168, Birkhäuser, Basel, 2006. [7] R. Dalmasso; Existence and uniqueness results for polyharmonic equations, Nonlinear Anal. TMA 36(1999), 131-137. [8] R. J. Duffin; On a question of Hadamard concerning super-biharmonic functions, J. Math. Phys., 27 (1949), 253-258. [9] E. N. Dancer; Global solution branches for positive mappings, Arch. Rat. Mech. Anal., 52(1973), 181-192. [10] P. R. Garabedian; A partial differential equation arising in conformal mapping, Pacific J. Math., 1(1951), 485-524. [11] F. Gazzola, H-Ch. Grunau, G. Sweers; Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics 1991, Berlin, Springer-Verlag, 2010. [12] D. Gilbarg, N. S. Trudinger; Elliptic partial differential equations of second order. Second edition. Springer-Verlag, Berlin, 1983. [13] H. C. Grunau, G. Sweers; Positivity for perturbations of polyharmonic operators with Dirich- let boundary conditions in two dimensions, Math. Nachr., 179(1996), 89-102. [14] H. C. Grunau, G. Sweers; Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann., 307(1997), 589-626. [15] H. C. Grunau, G. Sweers; Positivity properties of elliptic boundary value problems of higher order, Nonlinear Anal., 30(1997), 5251-5258. [16] J. Hadamard; Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, Mémoires présentés par divers savants à lÁcadémie des Sciences, 33(1908), 1-128. [17] J. Hadamard; Sur certains cas intéressants du problème biharmonique, Atti IVe Congr. In- tern. Mat. Rome, (1908), 12-14. [18] A. C. Lazer, P. J. McKenna; Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32(4) (1990), 537-578. [19] C. Loewner; On generation of solutions of the biharmonic equation in the plane by conformal mappings, Pacific J. Math., 3(1953), 417-436. [20] H. Mâagli, F. Toumi, M. Zribi; Existence of positive solutions for some polyharmonic non- linear boundary-value problems, Electron. J. Differential Equations, 58(2003), 19 pp. EJDE-2021/SI/01 BIHARMONIC EQUATIONS 253 [21] Ch. Pommerenke; Boundary behaviour of conformal maps, Berlin Heidelberg New York, Springer 1992. [22] H. S. Shapiro, M. Tegmark; An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36(1994), 99-101. [23] I. Sim, S. Tanaka; Three positive solutions for one-dimensional p-Laplacian problem with sign-changing weight, Appl. Math. Lett., 49(2015), 42-50. Ruyun Ma (corresponding author) School of Mathematics and Statistics, Xidian University, Xi’an 710071, China. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: ryma@xidian.edu.cn Zhongzi Zhao School of Mathematics and Statistics, Xidian University, Xi’an 710071, China Email address: 15193193403@163.com Dongliang Yan Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: yhululu@163.com 1. Introduction 2. Preliminaries 2.1. Principal eigenvalue 2.2. Shape of positive solutions 2.3. Global solutions branches for positive mappings 3. Main results 4. Bounds of solutions 4.1. A priori estimation 5. Rightward bifurcation 6. Direction turn of bifurcation 7. Second turn and proof of Theorem 3.1 Acknowledgements References