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.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y Competitivida

Malcher Figueiredo, Giovany de Jesus

Suárez Fernández, Antonio

English

idUS. Depósito de Investigación Universidad de Sevilla

Rev. Mat. Iberoam. 34 (2018), no. 2, 609–620doi 10.4171/rmi/997c© European Mathematical SocietySome remarks on the comparison principlein Kirchhoﬀ equationsGiovany M. Figueiredo and Antonio Sua´rezAbstract. In this paper we study the validity of the comparison princi-ple and the sub-supersolution method for Kirchhoﬀ type equations. Weshow that these principles do not work when the Kirchhoﬀ function isincreasing, contradicting some previous results. We give an alternativesub-supersolution method and apply it to some models.1. IntroductionIn the last years the nonlinear elliptic Kirchhoﬀ equation has attracted much at-tention, see for instance [12], [13], [15], [16], and references therein. The equationhas the following general form:(1.1){ −M(‖u‖2)Δu = f(x, u) in Ω,u = 0 on ∂Ω,where Ω ⊂ RN , N ≥ 1, is a bounded and regular domain,‖u‖2 :=∫Ω|∇u|2 dx,where M is 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 ﬁxedends 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 methodsMathematics Subject Classification (2010): 45M20, 35J25, 34B18..Keywords: Kirchhoﬀ equation, comparison principle, sub-supersolution method.610 G.M. Figueiredo and A. Sua´rezhave been used, mainly variational methods and ﬁxed point arguments, and alsobifurcation and sub-supersolution.In this note, we have two main objectives. On the one hand, we present someexamples demonstrating that some comparison results and those based on thesub-supersolution method appearing in the literature are not correct. On theother 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 2 we recall the previous resultsrelated to comparison and sub-supersolution method, and we present our mainresult. In Section 3 we give some counterexamples showing that some comparisonresults are not correct. Section 4 is devoted to prove our main result, and inSection 5 we apply our result to some speciﬁc examples.2. Previous and main resultsTo our knowledge, there are basically three results concerning the comparison andsub-supersolution results related to (1.1). Let us recall them. In [3] (Theorems 2and 3) the following result was proved:Theorem 2.1. Assume that:(M1) M is non-increasing in [0,+∞).(H) Define the functionH(t) := M(t2) t ,and assume that H is increasing and H(R) = R.Then:a) If there exist two non-negative functions u, u ∈ C2(Ω) such that u = u = 0on ∂Ω and(2.1) −M(‖u‖2)Δu ≤ −M(‖u‖2)Δu in Ω,then (comparison principle)u ≤ u in Ω.b) If(f1) f is increasing in the variable u for each x ∈ Ω fixed,and there exist two regular functions 0 ≤ u ≤ u in Ω, u = u = 0 on ∂Ωsatisfying(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 u of (1.1) such thatu ≤ u ≤ u in Ω.Comparison principle in Kirchhoff equations 611In [14] (Theorems 3.2 and 3.3), se also [8], the authors proved a similar resultto Theorem 2.1 in the case that M is increasing.Theorem 2.2. Assume that(M2) M is increasing.Then, the comparison principle holds. Moreover, if f satisfies (f1), the sub-supersolution method also works.Finally, in [4] the following result is shown:Theorem 2.3. Assume that M satisfies (M2) and(f2) f is a positive function.If there exist u ∈ W 1,∞(Ω), u ≥ 0 on ∂Ω, and a family (uδ) ⊂ W 1,∞0 (Ω) such that(2.3) −m0Δu ≥ f(x, u),‖uδ‖1,∞ → 0 as δ → 0, uδ ≤ u in Ω for δ small enough, and given α > 0, thereis δ0 such that(2.4) −Δuδ ≤1αf(x, uδ), for δ ≤ δ0,then there is a small enough δ > 0 such that there exists a solution u of (1.1) suchthat uδ ≤ u ≤ u in Ω.Of course, the above inequalities (2.3) and (2.4) are considered in the weaksense.Our main result reads as follows:Theorem 2.4. Assume (M0) and(M3) G(t) := M(t) t is invertible and denote by R(t) := G−1(t).Define now the non-local operatorR(w) := R( ∫Ωf(x,w)w dx).If there exist u, u ∈ H1(Ω) ∩ L∞(Ω) such that u ≤ u in Ω, u ≤ 0 ≤ u on ∂Ωsatisfying(2.5) −M(R(w))Δu ≥ f(x, u), −M(R(w))Δu ≤ f(x, u), ∀w ∈ [u, u],then there exists a solution u of (1.1) such that u ≤ u ≤ u in 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 w in the interval[u, u]. So, in order to verify for instance the ﬁrst inequality in (2.5), we have totake into account the sign of−Δu and the monotonicity of the map w → M(R(w)).See Section 5 for applications of this result.612 G.M. Figueiredo and A. Sua´rezIn Section 3, we show that Theorem 2.2 is not correct. Now, we deduce Theo-rems 2.1 b) and 2.3 from Theorem 2.4.Corollary 2.6. Assume that M is smooth, (M1), (H), (f1) and that there exista sub-supersolution in the sense of Theorem 2.1. Then, there exists a solution uof (1.1) such that u ≤ u ≤ u in Ω.Proof. Observe that if M is non-increasing and H is increasing, then G is increas-ing. Indeed, using that M ′ ≤ 0 we get thatG′(t) = M ′(t)t+M(t) ≥ 2tM ′(t) +M(t) = H ′(t1/2) > 0, t > 0,and then, w → R(w) is increasing because f is also increasing.Consider now that u, u is a sub-supersolution in the sense of Theorem 2.1. Weare going to show that it is also sub-supersolution in the sense of Theorem 2.4.We show this fact for u; for u we can apply an analogous reasoning. Since f isincreasing and 0 ≤ u ≤ u, we have thatM(‖u‖2)‖u‖2 ≤∫Ωf(x, u)u ≤∫Ωf(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 theexistence of a solution u of (1.1) and u ∈ [u, u]. Corollary 2.7. Assume (M2), (H), (f2) and that there exists a sub-supersolutionin the sense of Theorem 2.3. Then, there exists a solution u of (1.1) such thatu ≤ u ≤ u in Ω.Proof. Observe that if M is increasing, then G is increasing. Assume now the exis-tence of a supersolution u and family of sub-solution uδ in the sense of Theorem 2.3.Then,−Δu ≥ f(x, u)m0≥ f(x, u)M(R(w)) ∀w ∈ [u, u].Consider nowα = max0≤w≤uM(R(w)),and take u = uδ for some δ ≤ δ0 given by Theorem 2.3. Then, using that f ≥ 0,−Δu ≤ 1αf(x, u) ≤ 1M(R(w)) f(x, u) ∀w ∈ [u, u],and so, u, u is sub-supersolution in the sense of Theorem 2.4. Theorem 2.4 con-cludes the result. Comparison principle in Kirchhoff equations 6133. CounterexamplesIn this section we have two objectives: to show that when M satisﬁes only (M2),or M satisﬁes only (M1) (Cases 1 and 2 below, respectively), the comparisonprinciple and the sub-supersolutions fail.For that considerΩ = (0, π), u1 := sin(x), u2 := 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 = π2, ‖u2‖2 = π33,andmaxx∈[0,π]u1(x)u2(x)= maxx∈[0,π]sin(x)x(π − x) =4π2:= ρ∗  0.4083In order to prove that the comparison principle fails, we take u = ρu2, ρ > 0 andu = u1. So,−M(‖u‖2)Δu ≤ −M(‖u‖2)Δu for all x ∈ (0, π)if and only if(3.2) M(π/2) ≤ 2ρM(ρ2π33).Hence, if(3.3) ρ < ρ∗ and M(π/2) ≤ 2ρM(ρ2π33),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) in Ω,u = 0 on ∂Ω.Observe that the solutions of (3.4) areu0 = k sin(x),with k satisfying(3.5) kM(k2π2)= 1.614 G.M. Figueiredo and A. Sua´rezConsider the pair (u, u) = (ρu2, 0). Then there is a pair of sub-supersolutionof (3.4) in the sense of Theorem 2.2 provided that(3.6) 2ρM(ρ2π2)≥ 1.Then, if the method is applicable, then there exists a solution u0 of (3.4) such that0 ≤ u0 ≤ u = ρu2 in Ω.Hence, if(3.7) ρ < kρ∗ and 2ρM(ρ2π2)≥ 1,where k satisﬁes (3.5), then the sub-supersolution method fails.Case 1. M is increasing. Consider in this case c = 0, a > 0 and p > 0 in (3.1).Then, (3.3) is equivalent toa+ b(π2)p≤ 2ρ(a+ b(ρ2π33)p )and ρ < ρ∗.Taking b large, we need that for some ρ < ρ∗,121(π2 23)p < ρ1+2p.By continuity, it is enough that the above inequality holds for ρ = ρ∗, that is,121(π2 23)p <( 4π2)1+2p⇐⇒ 1 < 8π2( 323π2)p,which is true for p large.Now, we analyze the sub-supersolution method. First, observe that since M isincreasing, (3.4) possesses a unique solution, u0 = k sin(x), where k satisﬁes(3.8) k(a+ b(k2π2)p )= 1.Observe that if a → 0, thenk(a) →(1b)1/(2p+1)( 2π)p/(2p+1).Takep >ln(π2/8)ln(32/(3π2)).Then there exists a0 > 0 such that for all a ∈ (0, a0) we have(3.9)a+ b(16k23π)pa+ b(k2π2)p > π28.Comparison principle in Kirchhoff equations 615Hence, u is supersolution if (see (3.6))2ρ(a+ b(ρ2π33)p )≥ 1.By continuity, (3.7) is veriﬁed for some ρ < ρ∗ if24π2k(a+ b(( 4π2 k)2 π33)p )> 1,which is equivalent, using (3.8), to (3.9).Case 2. M is decreasing. Take in this case a, c > 0 and p < 0. In this case, (3.3)is equivalent to(3.10) a(1− 2ρ) ≤ b(2ρ(ρ2π33+ c)p−(π2+ c)p )and ρ < ρ∗.Take ρ2 < 32π2 . Then(3.11)π/2 + cρ2 π3/3 + c> 1.For this ρ ﬁxed, take p such that(3.12) 2ρ >( π/2 + cρ2 π3/3 + c)p,or equivalently,2ρ(ρ2π33+ c)p>(π2+ c)p.Now, take b large enough to have (3.10).Now, we show that the sub-supersolution method does not work for M onlysatisfying (M1). Take ρ and p such that (3.11) and (3.12) are satisﬁed. On theother hand, take b such that b(π/2 + c)p > 1, and then for all a > 0 we have(3.13) a+ b(π2+ c)p> 1.Then, thanks to (3.13), there exists at least a positive solution of (3.5), that is,k(a+ b(k2π2+ 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 ≥ 1a+ b cp.616 G.M. Figueiredo and A. Sua´rezIn order to verify (3.7) we need2ρ(a+ b(ρ2π33+ c)p )≥ 1 and ρ < k 4π2.Now, we prove that2ρ(a+ b(ρ2π33+ c)p )> a+ b(π2+ c)pwhence we conclude the result from (3.13). This is equivalent tob[2ρ(ρ2π33+ c)p−(π2+ c)p ]> a(1 − 2ρ),which is true taking a small and positive. Finally, we need to verify ρ < 4k/π2.But, observe that by (3.14), we have that, for a small, the above inequality is true.Remark 3.1. The above example shows that Theorem 3.3 in [14] (which assuresthat the sub-supersolution method and the comparison principle work), see alsoTheorem 2.3 in [11], seems not correct. Hence, the existence results of, for in-stance, [1], [2], [6], have been obtained using a method that fails.4. Proof of Theorem 2.4First, we are going to transform our equation (1.1) into another non-local ellipticequation. Indeed, multiplying (1.1) by u and integrating, we getM(‖u‖2)‖u‖2 =∫Ωf(x, u)u dx.By (H3), G is invertible, and so‖u‖2 = R(∫Ωf(x, u)u dx)= 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‖, and so itsuﬃces to apply Theorem 3.2 in [7]. This completes the proof.In the following result, we prove a speciﬁc comparison principle which is validwhen M only satisﬁes hypothesis (H) and the second member of the equation isconstant. Deﬁne e to be the unique positive solution of the equation(4.2){ −Δe = 1 in Ω,e = 0 on ∂Ω.Comparison principle in Kirchhoff equations 617Lemma 4.1. Assume that M satisfies (H) and let ui ∈ H10 (Ω), i = 1, 2, befunctions such that−M(‖ui‖2)Δui = fi ∈ R+and f1 ≤ f2. Then, u1 ≤ u2 in Ω.Proof. Observe that(4.3) M(‖ui‖2)ui = fi e,and then u1 ≤ u2 if and only if(4.4)f1M(‖u1‖2) ≤f2M(‖u2‖2) .But observe that, from (4.3),fi =M(‖ui‖2)‖ui‖‖e‖ ,and then (4.4) is equivalent to(4.5) ‖u1‖ ≤ ‖u2‖.Since M(‖ui‖2)‖ui‖ = fi‖e‖ and due to (H), (4.5) follows. Remark 4.2. We would like to emphasize that Lemma 4.1 is only true when fi,i = 1, 2 are real numbers.5. ApplicationsIn this section we apply our result to some models. We only assume that Msatisﬁes (M3). Denote by λ1 > 0 the principal eigenvalue of the Laplacian andϕ1 > 0 the eigenfunction associated to it with ‖ϕ1‖∞ = 1.Example 1. Consider the equation(5.1){ −M(‖u‖2)Δu = λuq in Ω,u = 0 on ∂Ω,where λ ∈ R and 0 < q < 1. This problem was analyzed in [3] when M satis-ﬁes (M1) and (H). We are going to show that (5.1) possesses a positive solution ifand only if λ > 0. From the maximum principle, if λ ≤ 0 problem (5.1) does nothave any positive solution. Assume λ > 0 and take as sub-supersolutions u = εϕ1and u = Ke with ε,K > 0 to be chosen. Then, u is supersolution ifK1−q ≥ 1m0λ ‖e‖q∞.618 G.M. Figueiredo and A. Sua´rezFix such K. Then, u is subsolution ifM(R(w)) ε1−q ≤ λλ1, ∀w ∈ [u, u].It is enough to take ε small such that the above inequality holds and that u ≤ u.Example 2. Consider the classical concave-convex equation(5.2){ −M(‖u‖2)Δu = λuq + up in Ω,u = 0 on ∂Ω,where λ ∈ R and 0 < q < 1 < p. Again we assume only that M satisﬁes (M3).We show that there exists at least a positive solution for λ small and positive. Forthat, again take the same sub-supersolution of the above example. We can showthat u = Ke es supersolution provided thatm0K1−q ≥ λ ‖e‖q∞ +Kp−q ‖e‖p∞.Then there exists λ0 > 0 such that, for λ ∈ (0, λ0), there exists K0 such thatu = K0e is supersolution.Now, u = εϕ1 is subsolution provided thatM(R(w)) ε1−q λ1 ≤ λ+ εp−q ϕp−q1 , ∀w ∈ [u, u].It suﬃces again to take ε small.Example 3. Consider now the logistic equation(5.3){ −M(‖u‖2)Δu = λu − up in Ω,u = 0 on ∂Ω,where λ ∈ R and 1 < p. In this example we assume, in addition to (M3), thatthere exist two positive constants m0,m∞ such that(5.4) m0 ≤ M ≤ m∞.This equation was studied in [5] when M = M(u) is a continuous function fromLp(Ω) into R that satisﬁes (5.4); however, they do not assume (M3). They used aﬁxed point argument and showed the existence of positive solution for λ > λ1m∞(Theorem 2.1 in [5]).We obtain a similar result for the Kirchhoﬀ equation (5.3) by the sub-supersolu-tion method. Observe that in this case we can not apply Theorem 2.1. Take u = λ.It is clear that u is supersolution. As subsolution, take u = εϕ1. Then we need thatM(R(w))λ1 + (εϕ1)p−1 ≤ λ, ∀w ∈ [u, u].Then there exists at least a positive solution for λ > λ1m∞.Remark 5.1. After we have ﬁnished and revised the paper, we have known the pa-per [10], where the authors give some counterexamples showing that equation (1.1)does not enjoy the comparison principle nor the sub supersolutions method.Acknowledgements. We thank the anonymous referee for carefully reading ofthis article and many useful comments.Comparison principle in Kirchhoff equations 619References[1] Afrouzi, G.A., Chung, N.Y. and Shakeri, S.: Existence of positive solutionsfor Kirchhoﬀ type equations. Electron. J. Diﬀerential Equations 180 (2013), 1–8.[2] Afrouzi, G.A., Chung, N.Y. and Shakeri, S.: Positive solutions for a semi-positone problem involving nonlocal operator. Rend. Sem. Mat. Univ. Padova 132(2014), 25–32.[3] Alves, C.O. and Correˆa, F. J. S. A.: On existence of solutions for a class ofproblem involving a nonlinear operator. Comm. Appl. Nonlinear Anal. 8 (2001),no. 2, 43–56.[4] Alves, C.O. and Correˆa, F. J. S. A.: A sub-supersolution approach for a quasi-linear Kirchhoﬀ equation. J. Math. Phys. 56 (2015), no. 5, 051501, 12 pp.[5] Chipot, M. and Correˆa, F. J. S.A.: Boundary layer solutions to functional ellipticequations. Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 381–393.[6] Chung, N.T.: An existence result for a class of Kirchhoﬀ type systems via sub andsupersolutions method. Appl. Math. Lett. 35 (2014), 95–101.[7] Correˆa, F. J. S.A., Delgado, M. and Sua´rez, A.: Some nonlinear heteroge-neous problems with nonlocal reaction term. Adv. Diﬀerential Equations 16 (2011),no. 7-8, 623–641.[8] Dai, G. and Ma, R.: Solutions for a p(x)-Kirchhoﬀ type equation with Neumannboundary data. Nonlinear Anal. Real World Appl. 12 (2011), no. 5, 2666–2680.[9] Figueiredo, G.M., Morales-Rodrigo, C., Santos, J. J. R. and Sua´rez, A.:Study of a nonlinear Kirchhoﬀ equation with non-homogeneous material. J. Math.Anal. Appl. 416 (2014), no. 2, 597–608.[10] Garc´ıa-Melia´n, J. and Iturriaga, L.: Some counterexamples related to the sta-tionary Kirchhoﬀ equation. Proc. Amer. Math. Soc. 144 (2016), no. 8, 3405–3411.[11] Han, X. and Dai, G.: On the sub-supersolution method for p(x)-Kirchhoﬀ typeequations. J. Inequal. Appl. 2012 (2012), 2012:283, 11pp.[12] Lei, C. -Y., Liao, J. -F., and Tang, C. -L.: Multiple positive solutions for Kirch-hoﬀ type of problems with singularity and critical exponents. J. Math. Anal. Appl.421 (2015), no. 1, 521–538.[13] Liang, Z., Li, F., and Shi, J.: Positive solutions to Kirchhoﬀ type equations withnonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincare´ Anal.Non Line´aire 31 (2014), no. 1, 155–167.[14] Ma, R., Dai, G. and Gao, C.: Existence and multiplicity of positive solutions fora class of p(x)-Kirchhoﬀ type equations. Bound. Value Probl. (2012), 2012:16.[15] Naimen, D.: The critical problem of Kirchhoﬀ type elliptic equations in dimensionfour. J. Diﬀerential Equations 257 (2014), no. 4, 1168–1193.[16] Sun, G., and Teng, K.: Existence and multiplicity of solutions for a class offractional Kirchhoﬀ-type problem. Math. Commun. 19 (2014), no. 1, 183–194.The authors have been partially supported for the following projects: CNPQ-Proc.400426/2013-7 (GF and AS), CNPQ-Proc. 301242/2011-9 (GF) and Ministerio de Economı´ay Competitividad under grants MTM2012-31304 and MTM2015-69875-P (AS).620 G.M. Figueiredo and A. Sua´rezReceived December 29, 2015.Giovany M. Figueiredo: Departamento de Matema´tica, Campus Universita´rioDarcy Ribeiro, Universidade de Bras´ılia, 70910-900, Bras´ılia DF, Brazil.E-mail: giovany unb@yahoo.comAntonio Sua´rez: Departamento de Ecuaciones Diferenciales y Ana´lisis Nume´rico,Facultad de Matema´ticas, Calle Tarﬁa s/n, 41012 Sevilla, Spain.E-mail: suarez@us.es

Some remarks on the comparison principle in Kirchhoff equations

https://idus.us.es/bitstream/handle/11441/76605/Some%20remarks%20on%20the%20comparison%20principle%20in%20Kirchhoff%20equations.pdf?sequence=1&isAllowed=y

Some remarks on the comparison principle in Kirchhoff equations

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla