Uniqueness theorem for p-biharmonic equations

Transcription

1 Electronic Journal of Differential Equations, Vol. 2222, No. 53, pp ISSN: URL: or ftp ejde.math.swt.edu login: ftp Uniqueness theorem for p-biharmonic equations Jiří Benedikt Abstract The goal of this paper is to prove existence and uniqueness of a solution of the initial value problem for the equation u p 2 u = λ u q 2 u where λ R and p, q > 1. We prove the existence for p q only, and give a counterexample which shows that for p < q there need not exist a global solution blow-up of the solution can occur. On the other hand, we prove the uniqueness for p q, and show that for p > q the uniqueness does not hold true we give a corresponding counterexample again. Moreover, we deal with continuous dependence of the solution on the initial conditions and parameters. 1 Introduction In 2, Drábek and Ôtani proved [3] that the initial value problem u t p 2 u t = λ ut p 2 ut, t [t, t + ε], ut = α, u t = β, u t p 2 u t = γ, u t p 2 u t t=t = δ 1.1 where λ > and p > 1, has a unique locally defined solution for some ε >. The equation in 1.1 is a generalization of the one-dimensional version of the well-known linear clamped plate equation, which we obtain choosing p = 2 in 1.1. It should be mentioned that the existence and uniqueness problem for 1.1 cannot be inferred from the classical theory. Indeed, let us denote v := u p 2 u and rewrite 1.1 as the equivalent problem u t = vt 2 p p 1 vt, ut = α, u t = β, v t = λ ut p 2 ut, vt = γ, v t = δ, t [t, t + ε]. 1.2 Mathematics Subject Classifications: 34A12, 34C11, 34L3. Key words: p-biharmonic operator, existence and uniqueness of solution, continuous dependence on initial conditions, jumping nonlinearity. c 22 Southwest Texas State University. Submitted April 15, 22. Published June 1, 22. 1

2 2 Uniqueness theorem for p-biharmonic equations EJDE 22/53 Whenever p 2, at least one of the right-hand sides in 1.2 satisfies neither Lipschitz see, e.g., [2] nor any other general condition that guarantees existence or uniqueness of a solution. For example, the very general Kamke s Theorem or its corollaries Nagumo s Rosenblatt s, Osgood s, Tonelli s Criterion, see, e.g., [4, pp ] cannot be used to prove the uniqueness here. In what follows, we show how the situation gets more complicated as we carry forward to more general problems than 1.1, namely to problems with different growth of the nonlinearity depending on u and on u non-homogeneous equation, jumping nonlinearity, non-constant coefficients. Let us consider the problem with a non-homogeneous equation u t p 2 u t = λ ut q 2 ut, t I, ut = α, u t = β, u t p 2 u t = γ, u t p 2 u t t=t = δ 1.3 where λ R, p, q > 1 and I = [t, t 1 ], t < t 1, or I = [t,. Taking p = q in 1.3 we obtain 1.1, but for p q the situation is more complex: for p < q we lose the existence of a globally defined solution we call this a global existence, and for p > q we lose the uniqueness of a locally defined solution we call this a local uniqueness. In Sections 3 and 4 we introduce the corresponding counterexamples. We can further generalize 1.3 adding the jumping nonlinearity to the righthand side: u t p 2 u t = µ ut q 1 2 u + t ν ut q2 2 u t, t I, ut = α, u t = β, u t p 2 u t = γ, u t p 2 u t t=t = δ 1.4 where p, q 1, q 2 > 1, µ, ν R, u + = max{u, } positive part of u and u = max{ u, } negative part of u. Putting q := q 1 = q 2 and λ := µ = ν into 1.4 we arrive at 1.3. Now the situation is analogous to the previous case 1.3: to prove the global existence we have to assume p max{q 1, q 2 }, and to prove the local uniqueness we must have p min{q 1, q 2 }. Taking into account non-constant coefficients in 1.3 we obtain: atu t p 2 u t = bt ut q 2 ut, t I, ut = α, u t = β, at u t p 2 u t = γ, at u t p 2 u t t=t = δ 1.5 where a, b CI and a >. When p > 2, it is not enough to assume p q for proving the local uniqueness. We have to add a condition on b. It suffices

3 EJDE 22/53 Jiří Benedikt 3 to assume b or b on the whole interval I, i.e., that b does not change its sign on I. Less restrictive is to assume that b have the property P stated below on I. Definiton 1.1 We say that a function f has a property P on the interval I = [t, t 1 ], or I = [t, ], if t I ξ >, ft t [ t, t + ξ] or ft t [ t, t + ξ] where I = [t, t 1, or I = [t,, respectively. In other words, for every point t of I except a contingent right boundary point there exists some right closed neighborhood of t in which f does not change its sign. Note that a continuous function that does not have the property P is, e.g., ft = t t sin 1 t t for t > t, ft =. It is clear that any constant function has the property P. We prove the both local and global existence and uniqueness for the most general non-homogeneous problem including the jumping nonlinearity and nonconstant coefficients as well: at u t p 2 u t = b1 t ut q1 2 u + t b 2 t ut q2 2 u t, t I, ut = α, u t = β, at u t p 2 u t = γ, at u t p 2 u t t=t = δ 1.6 where b 1, b 2 CI the other parameters are as above. Denoting u 1 := u and u 3 := a u p 2 u we can rewrite 1.6 as the equivalent initial value problem for a system of four equations of the first order u 1t = u 2 t, u 1 t = α, u 2t = a 1 p 1 t u3 t 2 p p 1 u3 t, u 2 t = β, u 3t = u 4 t, u 3 t = γ, u 4t = b 1 t u 1 t q1 2 u + 1 t b 2t u 1 t q2 2 u 1 t, u 4t = δ, The main results of this paper are the following. t I. 1.7 Proposition 1.2 local existence There exists ε > such that 1.6 has a solution on I = [t, t + ε]. Theorem 1.3 global existence Let p max{q 1, q 2 }. solution on I = [t,. Then 1.6 has a Corollary 1.4 If p max{q 1, q 2 }, then 1.4 has a solution on I = [t,. If p q, then 1.5 and 1.3 have a solution on I = [t,.

4 4 Uniqueness theorem for p-biharmonic equations EJDE 22/53 Proposition 1.5 local uniqueness Let one of these conditions hold true: α + β + γ + δ > or p min{q 1, q 2 }. Moreover, let at least one of the following conditions hold true: p 2 or α = β = or γ + δ > or there exists some right closed neighborhood of t in which neither b 1 nor b 2 changes its sign. Then there exists ε > such that 1.6 has at most one solution on I = [t, t + ε]. Remark 1.6 For the special cases 1.3 and 1.4 of 1.6 the last condition of the latter four is trivially satisfied, and so it remains to satisfy only one of the former two conditions. Theorem 1.7 global uniqueness Let p min{q 1, q 2 }. Further, let p 2 or functions b 1, b 2 have the property P on I see Definition 1.1. Then 1.6 has at most one solution. Corollary 1.8 If p min{q 1, q 2 } and, furthermore, p 2 or neither b 1 nor b 2 changes its sign on I, then 1.6 has at most one solution. If p q and, furthermore, p 2 or b has the property P on I, then 1.5 has at most one solution. If p q and, furthermore, p 2 or b does not change its sign on I, then 1.5 has at most one solution. If p min{q 1, q 2 }, then 1.4 has at most one solution. If p q, then 1.3 has at most one solution. The paper is organized as follows. In Section 2 we define the solution of 1.6. In Section 3 we prove Proposition 1.2 and Theorem 1.3. Section 4 contains a proof of Proposition 1.5 and Theorem 1.7. In Section 5 we introduce some open problems related to this paper. Tables 1 and 2 summarize the cases when the global existence, and the local uniqueness, respectively, of a solution of 1.3 is guaranteed or foreclosed there exists a counterexample. The following two corollaries are consequences of the global existence guaranteed by Theorem 1.3 and the global uniqueness guaranteed by Theorem 1.7. The reader is invited to accomplish their proofs following, e.g., that of [2, Th. 4.1, p. 59].

5 EJDE 22/53 Jiří Benedikt 5 p q YES Corollary 1.4 α, β, γ, δ, α + β + γ + δ > NO Example 3.1, or Remark 3.2 blow-up λ > α, β, γ, δ, α + β + γ + δ < to or α = β = γ = δ = YES trivial p < q κ 1, κ 2 {α, β, γ, δ} : κ 1 κ 2 <? λ = YES trivial λ <? Table 1: Existence of a solution of 1.3 on I = [t,. α + β + γ + δ > YES Proposition 1.5, Remark 1.6 p q YES Proposition 1.5, Remark 1.6 α = β = γ = δ = λ > NO Example 4.5 p > q λ = YES trivial λ <? Table 2: Uniqueness of a solution of 1.3 on I = [t, t + ε] for some ε >. Corollary 1.9 Let p min{ q 1, q 2 }. Further, let p 2 or b 1, b 2 have the property P on [a, b]. Let ũ be a solution of 1.7 with p = p, q 1 = q 1, q 2 = q 2, a = ã >, b 1 = b 1, b 2 = b 2, α = α, β = β, γ = γ, δ = δ, t = and I = [a, b], a < < b. Then there exists ε > such that for any p, q 1, q 2, α, β, γ, δ, R and a, b 1, b 2 CI satisfying p p + q 1 q 1 + q 2 q 2 + a ã CI + b 1 b 1 CI + b 2 b 2 CI + + α α + β β + γ γ + δ δ + < ε all solutions u = ut, p, q 1, q 2, a, b 1, b 2, α, β, γ, δ, of 1.7 with t = exist over I, and, as p, q 1, q 2, a, b 1, b 2, α, β, γ, δ, p, q 1, q 2, ã, b 1, b 2, α, β, γ, δ,, ut, p, q 1, q 2, a, b 1, b 2, α, β, γ, δ, ũt = ut, p, q 1, q 2, ã, b 1, b 2, α, β, γ, δ, uniformly over [a, b]. Corollary 1.1 Let p = q 1 = q 2. Further, let p 2 or b 1, b 2 have the property P on [a, b]. Let p, q 1, q 2, α, β, γ, δ, R, a < < b, and ã, b 1, b 2 CI, a >, be fixed. Then there exists a solution ũ of 1.7 with p = p, q 1 = q 1, q 2 = q 2, a = ã, b 1 = b 1, b 2 = b 2, α = α, β = β, γ = γ, δ = δ, t = and I = [a, b], and the conclusion of Corollary 1.9 holds true.

6 6 Uniqueness theorem for p-biharmonic equations EJDE 22/53 This paper is a brief version of the first chapter of the author s diploma thesis [1] which is available in Czech only. 2 Preliminaries Let us define a function ψ p : R R, p > 1, by ψ p s = s p 2 s for s, and ψ p =. Now we can rewrite 1.6 as atψp u t = b1 tψ q1 u + t b 2 tψ q2 u t, t I, ut = α, u t = β, at ψ p u t = γ, atψp u t t=t = δ. 2.1 We denote p = p p 1. One can simply show that ψ p and ψ p are inverse functions. The problem 1.7 then takes the form u 1t = u 2 t, u 1 t = α, u 2t = ctψ p u 3 t, u 2 t = β, u 3t = u 4 t, u 3 t = γ, u 4t = b 1 tψ q1 u + 1 t b 2tψ q2 u 1 t, u 4t = δ, t I 2.2 where ct = ψ p 1 at c CI, c >. Definiton 2.1 By a solution of 2.2 we understand a vector function u = u 1, u 2, u 3, u 4 of the class C 1 I 4 which satisfy the equations in 2.2 at every point of I, and fulfill the initial conditions in 2.2. By a solution of the problem 2.1 we understand a function u of the class C 2 I, such that u, u, aψ p u, aψ p u is a solution of the corresponding problem 2.2. Remark 2.2 We transferred the problem of existence and uniqueness of a solution of 2.1 i.e. 1.6 to the equivalent problem for Existence Proof of Proposition 1.2 By integration of the equations in 2.2 we obtain that u is a solution of 2.2 if and only if u 1, u 3 is a fixed point of the operator T : CI CI CI CI defined by T u, v = α + βt + γ + δt + t cψ p v d, t b 1 ψ q1 u + b 2 ψ q2 u d.

7 EJDE 22/53 Jiří Benedikt 7 The reader is invited to prove that there exists ε > such that the Schauder Fixed Point Theorem guarantees the existence of at least one fixed point of T. It completes the proof of Proposition 1.2 see Remark 2.2. Now we want to prove that the local solution can be extended to, i.e., that there exists a solution of 2.2 on I = [t,. We find that it is not always possible, and we must add some conditions on the parameters in 2.2. We begin with the example which shows that it is necessary. Example 3.1 Let in 1.3 p < q and λ >. Let H > t be arbitrary fixed. Then one can compute that the function ut = KH t r where r = 2p 2 p p 1q pp + q p 1 1/q p 2pq p q and K = p q λq p 2p is a solution of 1.3 with α = KH t r, β = KrH t r 1, γ = Krr 1 p 1 H t r 2p 1, δ = Krr 1 p 1 r 2p 1H t r 2p 1 1 on I = [t, t 1 ] for any t 1 t, H. However, this solution cannot be extended to I = [t, H] because ut as t H. This situation is called a blow-up of the solution. Remark 3.2 Using Example 3.1 one can prove that each of the conditions α, β, γ, δ, α + β + γ + δ >, p < q 1 and c, b 1 C > on [t,, and α, β, γ, δ, α + β + γ + δ <, p < q 2 and c, b 2 C > on [t, is sufficient for existence of H > t such that there is no solution of 1.6 on I = [t, H]. The idea of the proof is based on comparison of solutions of the initial value problem 1.6. Remark 3.3 Example 3.1 can be generalized for the initial value problem of the 2n th -order n N 1 n ψ p u n t n = λψq ut, t I, u i t = α i, ψp u n t i t=t = β i, i =,..., n where p < q and 1 n λ >. The solution is defined similarly as for 2.1. Let H > t be arbitrary fixed. The reader is invited to justify that the function ut = KH t r where K = n 1 k= r = np p q p 1 n 1 n kp + kq k= and 1 n λq p np npq kp n kq 1/q p

8 8 Uniqueness theorem for p-biharmonic equations EJDE 22/53 is a solution of 3.1 with some initial conditions on I = [t, t 1 ] for any t 1 t, H. As in Example 3.1, this solution cannot be extended to I = [t, H] because ut as t H. Proof of Theorem 1.3 Now we begin the proof of the existence of a solution of 1.6 on I = [t, assuming p max{q 1, q 2 }. It suffices to prove that there exists at least one solution of 2.2 see Remark 2.2 on I = [t, t 1 ] for any t 1 satisfying t < t 1. Let us have the auxiliary problem û 1t = û 2 t, û 2t = Cψ p û 3 t, û 3t = û 4 t, û 4t = Bψ p û 1 t, û 1 t = ˆα, û 2 t = ˆβ, û 3 t = ˆγ, û 4 t = ˆδ, t [t, t 1 ] 3.2 where b 1 t B, b 2 t B and ct C on [t, t 1 ]. The vector function û = û 1, û 2, û 3, û 4 where û 1 t = Ke rt t, û 2 t = Kre rt t, Kr 2 p 1 Kr û 3 t = e rp 1t t 2 p 1, û 4 t = rp 1e rp 1t t, C C K > is arbitrary and is a solution of 3.2 with p 1 2 1/2p r =, BC p 1 Kr 2 p 1 Kr 2 p 1 ˆα = K, ˆβ = Kr, ˆγ =, ˆδ = rp 1. C C We choose K big enough to have α < ˆα, β < ˆβ, γ < ˆγ and δ < ˆδ. We shall prove that for any solution u = u 1, u 2, u 3, u 4 of 2.2 on I = [t, t 1 ] u 1 t û 1 t, u 2 t û 2 t, u 3 t û 3 t, u 4 t û 4 t 3.3 for every t [t, t 1 ]. We have u 1 t < û 1 t, and so the set T = { t [t, t 1 ] : u 1 t û 1 t} is non-empty and closed, and there exists t m = max{ t [t, t 1 ] : [t, t] T }. We can assume K 1. Then for any t [t, t m ] we have û 1 t 1 and u 4t B u 1 t q 1 Bû 1 t q 1 Bû 1 t p 1 = û 4t

9 EJDE 22/53 Jiří Benedikt 9 where q = q 1 if u 1 t, and q = q 2 if u 1 t <. For any t [t, t m ] we now have u 4 t δ + u 4 d ˆδ + û 4 d = û 4 t, t u 3 t γ + t u 4 d ˆγ + Similarly we can show that for t [t, t m ] Thus t u 2t û 2t and u 2 t û 2 t. t û 4 d = û 3 t. m m u 1 t m α + u 1 d < ˆα + û 1 d = û 1 t m. t t This inequality would for t m < t 1 contradict with the maximality of t m, and so t m = t 1 and 3.3 is proved. Using the standard continuation arguments, the proof of Theorem 1.3 is completed. 4 Uniqueness In this section we prove the local uniqueness Proposition 1.5. We distinguish the cases a α + β + γ + δ > and b α = β = γ = δ =. a Here u 1 does not change its sign on some right neighborhood of t. Hence in this case it suffices to prove the uniqueness for the problem without the jumping nonlinearity, i.e The proof is divided into four parts: Lemma 4.1 for p 2, q 2, Lemma 4.2 for p 2, q < 2, Lemma 4.3 for p > 2, q 2 and Lemma 4.4 for p > 2, q < 2. b We assume p min{q 1, q 2 } here see Proposition 1.5. Lemma 4.7 deals with this case. Before Lemma 4.7 we introduce the example of non-uniqueness of a solution of 1.3 for α = β = γ = δ = and p > q. In all proofs in this section we denote by A, B, C > such constants that at A i.e. ct A 1 p, b 1 t B, b 2 t B i.e. bt B for b from 1.5 and ct C i.e. at C 1 p for every t I. We can also assume t =. According to Remark 2.2, we prove the assertions for 2.2. For Lemmata , formulated for 1.5, the fourth equation in 2.2 takes the form u 4t = btψ q ut. Lemma 4.1 Let α + β + γ + δ >, p 2 and q 2. Then there exists ε > such that 1.5 has at most one solution on I = [t, t + ε]. Proof Let u and v be solutions of the special case of 2.2, corresponding to 1.5. From the former two equations we conclude u 3 t v 3 t = at ψ p u 1t ψ p v 1 t,

10 1 Uniqueness theorem for p-biharmonic equations EJDE 22/53 and from the latter two equations we obtain u 3t v 3 t = bt ψ q u 1 t ψ q v 1 t, t I. Then at ψ p u 1t ψ p v 1 t = t b ψ q u 1 ψ q v 1 d. 4.1 There exists a constant K 1 > such that u 1t K 1 and v 1 t K 1 on I. Since p 2, ψ p ψ pk 1 for K 1, and so at ψ p u 1t ψ p v 1 t C 1 p u 1 t p 1K p 2 1 C 1 p u 1t v 1 t. ψ p d v 1 t 4.2 There exists a constant K 2 > such that u 1 K 2 and v 1 K 2 on I. Since q 2, ψ qσ ψ qk 2 for σ K 2. For I it yields ψ q u 1 ψ q v 1 = Using the estimate u 1 v 1 = we conclude u1 v 1 ψ qσ dσ q 1K q 2 2 u 1 v σu 1σ v 1 σ dσ 2 1 v 1 CI 4.4 t b ψ q u 1 ψ q v 1 d t 4 q 1K q 2 2 B 1 v 1 CI. 4.5 We combine 4.1, 4.2 and 4.5, take the maximum over t I, and get 1 v 1 CI ε 4 q 1K q 2 p 1K p 2 2 B u 1 C1 p 1 v 1 CI. 4.6 For ε > small enough this implies u 1 = v 1, and so u 3 = v 3. Since u 1 = v 1 and u 1 = v 1, it is then u 1 = v 1. Thus u = v. Lemma 4.2 Let α + β + γ + δ >, p 2 and q < 2. Then there exists ε > such that 1.5 has at most one solution on I = [t, t + ε]. Proof We distinguish the cases i α, ii α =, β, iii α = β =, γ and iv α = β = γ =, δ. i We proceed as in the proof of Lemma 4.1. The assumption u 1 = v 1 = α guarantees the existence of a constant K 2 > such that u 1 K 2

11 EJDE 22/53 Jiří Benedikt 11 and v 1 K 2 in [, ε] for ε > small enough. Since q < 2, ψ qσ ψ qk 2 for σ K 2. Hence 4.3 still holds true for all I, and we arrive again at 4.6. ii We modify again the proof of Lemma 4.1. Due to the assumptions α =, β, u1 β and v1 K 2 > such that u1 small enough. Thus u1 ψ q ψ q v1 β as +. Hence there exists a constant K 2 and v1 K 2 for all, ε] with ε > = u 1 v 1 Using 4.7 instead of 4.3 we get ψ qσ dσ 1 v 1 CI ε q+2 q 1K q 2 p 1K p 2 2 B u 1 C1 p q 1Kq 2 2 u 1 v v 1 CI. 4.8 iii We follow again the proof of Lemma 4.1. By the assumptions α = β =, γ, u cψ v1 p γ and cψ p γ as +. Thus, there exists a constant K 2 > such that u1 K 2 2 and v1 K 2 2 for every, ε] with ε > small enough. Then u1 ψ q 2 v1 ψ q = 2 u 1 2 v 1 2 Using 4.9 instead of 4.3 we get ψ qσ dσ 1 v 1 CI ε 2q q 1K q 2 p 1K p 2 2 B u 1 C1 p q 1Kq u 1 v v 1 CI. 4.1 iv Here we cannot follow the proof of Lemma 4.1. Like 4.1, we can derive that for every t I btu1 t v 1 t bt t c ψ p u 3 ψ p v 3 d By the assumptions γ =, δ, u 1 p 1 cψ p δ as +. So there exists a constant K 1 > such that u 1 p 1 K 1 for any, ε] with ε > small enough. Thus, for every t I u 1 t = t u 1 d t K 1 p 1 d = K 1t p +1 p p + 1, i.e. u1t ψq t p +1 K K1 := ψ 1 q p p +1 >, and analogously ψq v1t t p +1 K 1 for all t, ε]. Since q < 2, ψ q σ ψ q K 1 for σ K 1. Thus, for all t, ε], btu 1 t v 1 t = bt t p +1 ψ q ψ q u1 t t p +1 ψ q ψ q v1 t t p +1 =

12 12 Uniqueness theorem for p-biharmonic equations EJDE 22/53 = bt t p +1 u1 t ψq t p +1 ψ q v1 t t p +1 ψ q σ dσ t 2 qp +1 q 1 K q 2 1 u 3t v 3 t Since u3 δ and v3 δ as +, there exists K 2 > such that u3 K2 and v3 K2 for any, ε]. Since p 2, ψ p σ ψ p K 2 for σ K 2, and so ψ p u3 ψ p v3 Since, analogously to 4.4, = u 3 v 3 ψ p σ dσ p 1K p 2 2 u 3 v u 3 v v 3 CI I, 4.14 we have the following estimate for all t I: u3 v3 bt t c ψ p ψ p ψ p d t p +2 p 1K p 2 2 BC 3 v 3 CI Putting 4.12, 4.11 and 4.15 together and passing to the maximum over I for t = the inequality is trivially satisfied we obtain 3 v 3 CI ε q 1p +1+1 p 1 q 1 2 q K 1 K p 2 2 BC 3 v 3 CI Since for any p, q > 1 we have q 1p >, the proof is complete. Lemma 4.3 Let α + β + γ + δ >, p > 2 and q 2. Moreover, let γ + δ > or b not change its sign on some right closed neighborhood of t. Then there exists ε > such that 1.5 has at most one solution on I = [t, t + ε]. Proof We distinguish the cases i γ, ii γ =, δ, iii γ = δ =, α and iv γ = δ = α =, β. Let again u a v be solutions of the special case 2.2, corresponding to 1.5. i Since u 1 = v 1 = cψ p γ, there exists a constant K 1 > such that u 1t K 1 and v 1 t K 1 for all t [, ε] with ε > small enough. We have p > 2, and so ψ p ψ pk 1 for K 1. Hence 4.2 holds true, and we get 4.6. ii As in the part iv of the proof of Lemma 4.2, there exists a constant K 1 > such that u 1 t K1 and v 1 t K1 for all t, ε] with ε > t p 1 t p 1 small enough. Hence at ψ p u 1t ψ p v 1 ψp u t = at t C 1 p t u 1 t t p 1 v 1 t t p 1 1 t t p 1 ψ p v 1 t t p 1 ψ pσ dσ t 2 p p 1K p 2 1 C 1 p u 1t v 1 t. 4.17

13 EJDE 22/53 Jiří Benedikt 13 Using 4.17 instead of 4.2 we obtain iii We can assume 1 v 1 CI ε p +2 q 1K q 2 p 1K p 2 ft := 2 B u 1 C1 p t b d > t, ε] 1 v 1 CI otherwise b = for all [, t ] with some t >, and the uniqueness is then trivial. Since u 1 = α, there exists a constant K 1 > such that u 1 K 1, and so u 3 K q 1 1 b for all [, ε] with ε > small enough. We suppose that b and u 3 does not change its sign on I. Hence for any t I Thus u 3 t = t u 3 d K q 1 1 t b d = K q 1 1 ft. u 1t = ctψ p u 3 t K q 1p 1 1 A 1 p f p 1 t, and the same estimate holds for v 1 t, t I. For t, ε] we can write at ψ p u 1t ψ p v 1 t C 1 p u 1 ft ψ t v 1 p ψ t f p 1 t p f p 1 t = = C 1 p ft u 1 t f p 1 t v 1 t f p 1 t ψ p d 4.19 q 1p 2 p 1 p 1K1 A p 2 p 1 C 1 p f 2 p t u 1t v 1 t. Using 4.3 and 4.4 we get for t I t b ψ q u 1 ψ q v 1 d t 2 q 1K q 2 2 ft 1 v 1 CI. 4.2 Obviously ft t 2 B. Putting 4.19, 4.1 and 4.2 together and passing to the maximum over I we arrive at 1 v 1 CI ε 2p q 1Kq 2 p 1K 2 B p 1 q 1p 2 p 1 1 A p 2 p 1 C 1 p iv We proceed as analogically to iii. We can assume ft := t q 1 b d > t, ε]. 1 v 1 CI.

14 14 Uniqueness theorem for p-biharmonic equations EJDE 22/53 Since u1 β, there exists a constant K 1 > such that u1 K1, and also u 3 K 1 q 1 b for [, ε] with ε > small enough. We suppose that b and u 3 does not change sign on I. For any t I we have u 3 t = t u 3 d K q 1 1 Analogously as 4.19 we can now for t, ε] derive at ψ p u 1t ψ p v 1 t t q 1 b d = K q 1 1 ft q 1p 2 p 1 p 1K1 A p 2 p 1 C 1 p f 2 p t u 1t v 1 t. There exists a constant K 2 > such that u1 K2 and v1 K2 for all, ε]. Since q 2, ψ qσ ψ qk 2 for σ K 2. Thus, 4.7 holds true for all t, ε]. Together with 4.4 it yields for t I t b ψ q u 1 ψ q v 1 d tq 1K q 2 2 ft 1 v 1 CI Obviously ft t q+1 B. Using 4.21, 4.1, 4.22 we obtain for the maximum over I 1 v 1 CI ε q+1p 1+1 q 1K q 2 p 1K 2 B p 1 q 1p 2 p 1 1 A p 2 p 1 C 1 p 1 v 1 CI. Since for p, q > 1 it is q + 1p >, we proved the assertion of this lemma. Lemma 4.4 Let α + β + γ + δ >, p > 2 and q < 2. Moreover, let γ + δ > or b not change its sign on some right closed neighborhood of t. Then there exists ε > such that 1.5 has at most one solution on I = [t, t + ε]. Proof We combine the previous techniques. Consequently, we distinguish the cases i α, γ, ii α, γ =, δ, iii α, γ = δ =, iv α =, β, γ, v α =, β, γ =, δ, vi α =, β, γ = δ =, vii α = β =, γ and viii α = β = γ =, δ. i As in the part i of the proof of Lemma 4.3, we can use 4.2, and, as in the part i of the proof of Lemma 4.2, we can use 4.3. Using 4.1 we arrive at 4.6. ii From 4.17, 4.1 and 4.3 we derive iii As 4.3 holds true, we follow the part iii of the proof of Lemma 4.3. iv From 4.2, 4.1 and 4.7 we get 4.8. v Here 4.17, 4.1 and 4.7 yield 1 v 1 CI ε p +q q 1K q 2 p 1K p 2 2 B u 1 C1 p 1 v 1 CI.

15 EJDE 22/53 Jiří Benedikt 15 vi Since 4.7 holds true for some K 2 >, we can proceed as in the part iv of the proof of Lemma 4.3. vii From 4.2, 4.1 and 4.9 we conclude 4.1. viii We follow the part iv of the proof of Lemma 4.2. Due to the assumptions γ =, δ, there exists a constant K 2 > such that u3 K2 and u3 K 2 for all, ε] with ε > small enough. Since p > 2, ψ p σ ψ p K 2 for σ K 2, and so 4.13 holds true. Now we put 4.12, 4.11 and 4.13 together to obtain As we promised, we show now that if p > q, α = β = γ = δ = and λ >, then 1.3 has a non-trivial solution besides of the trivial one, and so the uniqueness is broken. Example 4.5 Let in 1.3 p > q, α = β = γ = δ =, λ > and I = [, ε] with some ε >. Then one can compute that ut = and ut = Kt t r where r = 2p p q and K = 2 p p 1q pp + q p 1 2pq p q λp q 2p 1/q p are solutions of 1.3. Remark 4.6 Example 4.5 can be generalized cf. Example 3.1 for the initial value problem 3.1 of the 2n th -order, n N, where p > q, 1 n λ > and α i = β i =, i =,..., n 1. The reader is invited to justify that ut = and ut = KH t r where K = n 1 k= r = np p q p 1 n 1 n kp + kq k= and 1 n λp q np npq kp n kq 1/q p are solutions of 3.1. Lemma 4.7 Let α = β = γ = δ = and p min{q 1, q 2 }. Then there exists ε > such that 1.6 has at most one solution on I = [t, t + ε]. Proof Let u be a solution of 1.6. We prove that u 1 = u 2 = u 3 = u 4 = on [, ε] for some ε >. For t I atψ p u 1t t b 1 ψ q1 u b 2 ψ q2 u 1 d 4.23

16 16 Uniqueness theorem for p-biharmonic equations EJDE 22/53 Since for I obviously u 2 CI, we have B b 1 ψ q1 u b 2 ψ q2 u 1 B 2q1 2 1 q1 1 CI + 2q2 2 1 q2 1 CI 2q1 2 1 q1 p CI + 2q2 2 1 q2 p CI 1 p 1 CI 4.24 where I = [, ε ] with ε > arbitrary, but fixed, and ε ε. We used the assumption p min{q 1, q 2 } which implies that 1 qi p C[,ε], i = 1, 2, are increasing functions of ε. Using the estimate atψ p u 1t C 1 p u 1t p 1 we can infer from 4.23 and 4.24 that for every t I C 1 p u 1t p 1 B t 2q1 1 q1 p CI + 1 q2 p t2q2 CI 1 p 1 CI. Now we pass to the maximum for t I. If we suppose that ε 1, we obtain 1 p 1 CI ε2 min{q1,q2} BC p 1 q1 p CI + u q2 p CI 1 p 1 CI. For ε > small enough this inequality guarantees that p 1 CI =, and so u 1 =, u 1 =, and also u 2 = u 3 = u 4 = on I = [, ε]. Now that we completed the proof of Proposition 1.5. Theorem 1.7 is a direct consequence of this proposition. 5 Open Problems The main problems we leave open are: 1. Does the conclusion of Proposition 1.5 local uniqueness hold true even without the latter four conditions, i.e. for p > 2, γ = δ =, α or β nonzero and b 1 or b 2 changing its sign on arbitrarily small right neighborhood of t? If it did, then the sufficient condition for the global uniqueness see Theorem 1.7 would be p min{q 1, q 2 } only we showed that this assumption cannot be left out. We can simplify this problem: Can there exist two different solutions of u t = ψ p vt, ut = α, u t = β, v t = btψ p ut, vt =, v t =, t I 5.1 with I = [t, t 1 ], b CI arbitrary, p > 2 and α + β >? Note that the system of equations in 5.1 is homogeneous! 2. We gave Example 3.1 which showed that for p < q and some initial conditions the solution of 1.3 did not have to exist on [t,. In this Example we assumed λ >, for λ = the global existence is trivial, but for λ < we leave the question of global existence open see Table 1.

17 EJDE 22/53 Jiří Benedikt Analogously to the previous open problem, for λ <, α = β = γ = δ = and p > q we gave neither the proof of the local uniqueness of the solution of 1.3 nor a counterexample see Table 2, and so we leave it as an open question, too. Acknowledgements The author is partially supported by grant 21//376 from the Grant Agency of the Czech Republic and by grant F971/22-G4 from the Ministry of Education of the Czech Republic. References [1] Benedikt, J., Sturmova Liouvilleova úloha pro p biharmonický operátor Sturm Liouville Problem for p-biharmonic Operator, diploma thesis, Faculty of Applied Sciences, University of West Bohemia, Plzeň 21 in Czech. [2] Coddington, E. A., Levinson, N., Theory of Ordinary Differential Equations, McGraw Hill Book Company, Inc., New York Toronto London [3] Drábek, P., Ôtani, M., Global Bifurcation Result for the p-biharmonic Operator, Electron. J. Diff. Eqns., Vol. 2121, No. 48, pp [4] Hartman, P., Ordinary Differential Equations, John Wiley & Sons, Inc., Baltimore Jiří Benedikt Centre of Applied Mathematics University of West Bohemia Univerzitní 22, Plzeň Czech Republic benedikt@kma.zcu.cz Addendum: July 28, 23. It was brought to my knowledge by a colleague that in Remark 4.6, fourth line page 15 there should be ut = Kt t r instead of ut = KH t r. Even though I think that this mistake is not misleading for the reader, I want to correct it.