Some remarks on the comparison principle in Kirchhoff equations

Rev. Mat. Iberoam. 34 (2018), no. 2, 609–620 doi 10.4171/rmi/997 c European Mathematical Society Some remarks on the comparison principle in Kirchhoff equations Giovany M. Figueiredo and Antonio Su´arez Abstract. In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models. 1. Introduction In the last years the nonlinear elliptic Kirchhoff equation has attracted much attention, see for instance [12], [13], [15], [16], and references therein. The equation has the following general form: (1.1) −M(u2)Δu=f(x, u)inΩ, u=0 on∂Ω, where Ω ⊂RN,N≥1, is a bounded and regular domain, u2:= Ω |∇u|2dx, where Mis a continuous function verifying (M0)M:[0,+∞)→ [0,+∞) and ∃m0>0 such that M(t)≥m0>0∀t∈[0,+∞), and where f∈C(Ω×R). We assume (M0) along the paper. Problem (1.1) models small vertical vibrations of an elastic string with fixed ends when the density of the material is homogeneous and there is a external force, see [9] for a explication of the model. To study this problem different methods Mathematics Subject Classification (2010): 45M20, 35J25, 34B18.. Keywords: Kirchhoff equation, comparison principle, sub-supersolution method. 610 G. M. Figueiredo and A. Su´ arez have been used, mainly variational methods and fixed point arguments, and also bifurcation and sub-supersolution. In this note, we have two main objectives. On the one hand, we present some examples demonstrating that some comparison results and those based on the sub-supersolution method appearing in the literature are not correct. On the other hand, we prove a sub-supersolution method that includes the above ones, specifically those in [3], [4], and [14]. An outline of the paper is as follows: in Section 2we recall the previous results related to comparison and sub-supersolution method, and we present our main result. In Section 3we give some counterexamples showing that some comparison results are not correct. Section 4is devoted to prove our main result, and in Section 5we apply our result to some specific examples. 2. Previous and main results To our knowledge, there are basically three results concerning the comparison and sub-supersolution results related to (1.1). Let us recall them. In [3](Theorems2 and 3) the following result was proved: Theorem 2.1. Assume that: (M1)Mis non-increasing in [0,+∞). (H)Define the function H(t):=M(t2)t, and assume that His increasing and H(R)=R. Then: a) If there exist two non-negative functions u, u ∈C2(Ω) such that u=u=0 on ∂Ωand (2.1) −M(u2)Δu≤−M(u2)Δuin Ω, then (comparison principle) u≤uin Ω. b) If (f1)fis increasing in the variable ufor each x∈Ωfixed, and there exist two regular functions 0≤u≤uin Ω,u=u=0on ∂Ω satisfying (2.2) −M(u2)Δu≥f(x, u),−M(u2)Δu≤f(x, u),in Ω. then (sub-supersolution method)there exists a solution uof (1.1)such that u≤u≤uin Ω. Comparison principle in Kirchhoff equations 611 In [14] (Theorems 3.2 and 3.3), se also [8], the authors proved a similar result to Theorem 2.1 in the case that Mis increasing. Theorem 2.2. Assume that (M2)Mis increasing. Then, the comparison principle holds. Moreover, if fsatisfies (f1),thesubsupersolution method also works. Finally, in [4] the following result is shown: Theorem 2.3. Assume that Msatisfies (M2)and (f2)fis a positive function. If there exist u∈W1,∞(Ω),u≥0on ∂Ω,andafamily(uδ)⊂W1,∞ 0(Ω) such that (2.3) −m0Δu≥f(x, u), uδ1,∞→0as δ→0,uδ≤uin Ωfor δsmall enough, and given α>0,there is δ0such that (2.4) −Δuδ≤1 αf(x, uδ),for δ≤δ0, then there is a small enough δ>0such that there exists a solution uof (1.1)such that uδ≤u≤uin Ω. Of course, the above inequalities (2.3)and(2.4) are considered in the weak sense. Our main result reads as follows: Theorem 2.4. Assume (M0)and (M3)G(t):=M(t)tis invertible and denote by R(t):=G−1(t). Define now the non-local operator R(w):=RΩ f(x, w)wdx . If there exist u, u ∈H1(Ω) ∩L∞(Ω) such that u≤uin Ω,u≤0≤uon ∂Ω satisfying (2.5) −M(R(w))Δu≥f(x, u),−M(R(w))Δu≤f(x, u),∀w∈[u, u], then there exists a solution uof (1.1)such that u≤u≤uin Ω. Remark 2.5. (1) We would like to remark that Theorem 2.4 does not assume (f1). (2) Observe that condition (2.5) involves all the functions win the interval [u, u]. So, in order to verify for instance the first inequality in (2.5), we have to take into account the sign of −Δuand the monotonicity of the map w→ M(R(w)). See Section 5for applications of this result. 612 G. M. Figueiredo and A. Su´ arez In Section 3, we show that Theorem 2.2 is not correct. Now, we deduce Theorems 2.1 b) and 2.3 from Theorem 2.4. Corollary 2.6. Assume that Mis smooth, (M1),(H),(f1)and that there exist a sub-supersolution in the sense of Theorem 2.1. Then, there exists a solution u of (1.1)such that u≤u≤uin Ω. Proof. Observe that if Mis non-increasing and His increasing, then Gis increasing. Indeed, using that M≤0wegetthat G(t)=M(t)t+M(t)≥2tM(t)+M(t)=H(t1/2)>0,t>0, and then, w→R(w) is increasing because fis also increasing. Consider now that u,uis a sub-supersolution in the sense of Theorem 2.1.We are going to show that it is also sub-supersolution in the sense of Theorem 2.4. We show this fact for u;foruwe can apply an analogous reasoning. Since fis increasing and 0 ≤u≤u,wehavethat M(u2)u2≤Ω f(x, u)u≤Ω f(x, w)w∀w∈[u, u], and so, u2≤R(w)=⇒M(u2)≥M(R(w)) ∀w∈[u, u]. Hence, −Δu≤f(x, u) M(u2)≤f(x, u) M(R(w)) ∀w∈[u, u]. Then, (u, u) satisfies the hypotheses of Theorem 2.4 and we can conclude the existence of a solution uof (1.1)andu∈[u, u]. 2 Corollary 2.7. Assume (M2),(H),(f2)and that there exists a sub-supersolution in the sense of Theorem 2.3. Then, there exists a solution uof (1.1)such that u≤u≤uin Ω. Proof. Observe that if Mis increasing, then Gis increasing. Assume now the existence of a supersolution uand family of sub-solution uδin the sense of Theorem 2.3. Then, −Δu≥f(x, u) m0 ≥f(x, u) M(R(w)) ∀w∈[u, u]. Consider now α=max 0≤w≤uM(R(w)), and take u=uδfor some δ≤δ0given by Theorem 2.3. Then, using that f≥0, −Δu≤1 αf(x, u)≤1 M(R(w)) f(x, u)∀w∈[u, u], and so, u,uis sub-supersolution in the sense of Theorem 2.4.Theorem2.4 concludes the result. 2 Comparison principle in Kirchhoff equations 613 3. Counterexamples In this section we have two objectives: to show that when Msatisfies only (M2), or Msatisfies only (M1) (Cases 1 and 2 below, respectively), the comparison principle and the sub-supersolutions fail. For that consider Ω=(0,π),u 1:= sin(x),u 2:= x(π−x), and (3.1) M(t):=a+b(t+c)p,c≥0,a,b>0,p∈R. Observe that −Δu1=u1,−Δu2=2,u12=π 2,u22=π3 3, and max x∈[0,π] u1(x) u2(x)=max x∈[0,π] sin(x) x(π−x)=4 π2:= ρ∗≃0.4083 In order to prove that the comparison principle fails, we take u=ρu2,ρ>0and u=u1.So, −M(u2)Δu≤−M(u2)Δufor all x∈(0,π) if and only if (3.2) M(π/2) ≤2ρMρ2π3 3. Hence, if (3.3) ρ<ρ ∗and M(π/2) ≤2ρMρ2π3 3, then the comparison principle fails. To prove that the sub-supersolutions method fails, we consider the problem (3.4) −M(u2)Δu=f(x)=sin(x)inΩ, u=0 on∂Ω. Observe that the solutions of (3.4)are u0=ksin(x), with ksatisfying (3.5) kM k2π 2=1. 614 G. M. Figueiredo and A. Su´ arez Consider the pair (u, u)=(ρu2,0). Then there is a pair of sub-supersolution of (3.4) in the sense of Theorem 2.2 provided that (3.6) 2ρMρ2π 2≥1. Then, if the method is applicable, then there exists a solution u0of (3.4) such that 0≤u0≤u=ρu2in Ω. Hence, if (3.7) ρ<kρ ∗and 2ρM ρ2π 2≥1, where ksatisfies (3.5), then the sub-supersolution method fails. Case 1.Mis increasing. Consider in this case c=0,a>0andp>0in(3.1). Then, (3.3) is equivalent to a+bπ 2p≤2ρa+bρ2π3 3pand ρ<ρ ∗. Taking blarge, we need that for some ρ<ρ ∗, 1 2 1 π22 3p<ρ 1+2p. By continuity, it is enough that the above inequality holds for ρ=ρ∗,thatis, 1 2 1 π22 3p<4 π21+2p⇐⇒ 1<8 π232 3π2p, which is true for plarge. Now, we analyze the sub-supersolution method. First, observe that since Mis increasing, (3.4) possesses a unique solution, u0=ksin(x), where ksatisfies (3.8) ka+bk2π 2p=1. Observe that if a→0, then k(a)→1 b1/(2p+1)2 πp/(2p+1). Take p> ln(π2/8) ln(32/(3π2)). Then there exists a0>0 such that for all a∈(0,a 0)wehave (3.9) a+b16k2 3πp a+bk2π 2p>π2 8. Comparison principle in Kirchhoff equations 615 Hence, uis supersolution if (see (3.6)) 2ρa+bρ2π3 3p≥1. By continuity, (3.7) is verified for some ρ<ρ ∗if 24 π2ka+b(4 π2k)2π3 3p>1, which is equivalent, using (3.8), to (3.9). Case 2.Mis decreasing. Take in this case a, c > 0andp<0. In this case, (3.3) is equivalent to (3.10) a(1 −2ρ)≤b2ρρ2π3 3+cp−π 2+cpand ρ<ρ ∗. Take ρ2<3 2π2.Then (3.11) π/2+c ρ2π3/3+c>1. For this ρfixed, take psuch that (3.12) 2ρ>π/2+c ρ2π3/3+cp, or equivalently, 2ρρ2π3 3+cp>π 2+cp. Now, take blarge enough to have (3.10). Now, we show that the sub-supersolution method does not work for Monly satisfying (M1). Take ρand psuch that (3.11)and(3.12) are satisfied. On the other hand, take bsuch that b(π/2+c)p>1, and then for all a>0wehave (3.13) a+bπ 2+cp>1. Then, thanks to (3.13), there exists at least a positive solution of (3.5), that is, ka+bk2π 2+cp=1, and then, there exists at least a positive solution of (3.4). Observe that in this case, we can not assure that (3.4) has a unique solution. On the other hand, since p<0, observe that (3.14) k≥1 a+bc p. 616 G. M. Figueiredo and A. Su´ arez In order to verify (3.7) we need 2ρa+bρ2π3 3+cp≥1andρ<k 4 π2. Now, we prove that 2ρa+bρ2π3 3+cp>a+bπ 2+cp whence we conclude the result from (3.13). This is equivalent to b2ρρ2π3 3+cp−π 2+cp>a(1 −2ρ), which is true taking asmall and positive. Finally, we need to verify ρ<4k/π2. But, observe that by (3.14), we have that, for asmall, the above inequality is true. Remark 3.1. The above example shows that Theorem 3.3 in [14] (which assures that the sub-supersolution method and the comparison principle work), see also Theorem 2.3 in [11], seems not correct. Hence, the existence results of, for instance, [1], [2], [6], have been obtained using a method that fails. 4. Proof of Theorem 2.4 First, we are going to transform our equation (1.1) into another non-local elliptic equation. Indeed, multiplying (1.1)byuand integrating, we get M(u2)u2=Ω f(x, u)udx. By (H3), Gis invertible, and so u2=RΩ f(x, u)udx =R(u). Then, (1.1) is equivalent to the problem (4.1) ⎧ ⎨ ⎩ −Δu=f(x, u) M(R(u)) in Ω, u=0 on∂Ω. Observe that (4.1) is a non-local elliptic equation, without terms in u,andsoit suffices to apply Theorem 3.2 in [7]. This completes the proof. In the following result, we prove a specific comparison principle which is valid when Monly satisfies hypothesis (H) and the second member of the equation is constant. Define eto be the unique positive solution of the equation (4.2) −Δe=1 inΩ, e=0 on∂Ω. Comparison principle in Kirchhoff equations 617 Lemma 4.1. Assume that Msatisfies (H)and let ui∈H1 0(Ω),i=1,2,be functions such that −M(ui2)Δui=fi∈R+ and f1≤f2.Then,u1≤u2in Ω. Proof. Observe that (4.3) M(ui2)ui=fie, and then u1≤u2if and only if (4.4) f1 M(u12)≤f2 M(u22). But observe that, from (4.3), fi=M(ui2)ui e, andthen(4.4)isequivalentto (4.5) u1≤u2. Since M(ui2)ui=fieand due to (H), (4.5) follows. 2 Remark 4.2. We would like to emphasize that Lemma 4.1 is only true when fi, i=1,2 are real numbers. 5. Applications In this section we apply our result to some models. We only assume that M satisfies (M3). Denote by λ1>0 the principal eigenvalue of the Laplacian and ϕ1>0 the eigenfunction associated to it with ϕ1∞=1. Example 1.Consider the equation (5.1) −M(u2)Δu=λuqin Ω, u=0 on∂Ω, where λ∈Rand 0 <q<1. This problem was analyzed in [3]whenMsatisfies (M1)and(H). We are going to show that (5.1) possesses a positive solution if and only if λ>0. From the maximum principle, if λ≤0problem(5.1)doesnot have any positive solution. Assume λ>0 and take as sub-supersolutions u=εϕ1 and u=Ke with ε, K > 0 to be chosen. Then, uis supersolution if K1−q≥1 m0 λeq ∞.