Marini, Mauro

Matucci, Serena

Řehák, Pavel

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 10Mauro Marini; Serena Matucci; Pavel ŘehákZero convergent solutions for a class of p-Laplacian systemsIn: Jaromír Kuben and Jaromír Vosmanský (eds.): Equadiff 10, CzechoslovakInternational Conference on Differential Equations and Their Applications, Prague,August 27-31, 2001, [Part 2] Papers. Masaryk University, Brno, 2002. CD-ROM; alimited number of printed issues has been issued. pp. 279--287.Persistent URL: http://dml.cz/dmlcz/700360Terms of use:© Institute of Mathematics AS CR, 2002Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czEquadiff 10, August 27–31, 2001Prague, Czech RepublicEquadiff 10 CD ROMPapers, pp. 279–287Zero Convergent Solutions for a Class ofp-Laplacian SystemsMauro Marini1, Serena Matucci2 and Pavel Řehák31 Department of Electronic and Telecommunications, Faculty of Engineering,University of Florence, via S. Marta 3, I-50139 Florence, ItalyEmail: marini@ing.unifi.it2 Department of Electronic and Telecommunications, Faculty of Engineering,University of Florence, via S. Marta 3, I-50139 Florence, ItalyEmail: matucci@det.unifi.it3 Mathematical Institute, Academy of Sciences of Czech Republic,Žižkova 22, CZ-616 62 Brno, Czech RepublicEmail: rehak@math.muni.czAbstract. We present some results about the existence of positive decay-ing solutions for a class of systems of two second order coupled nonlin-ear equations with p-laplacian operator, p > 1. In addition the general-ized Emden-Fowler type systems are considered and necessary and suffi-cient conditions are given in order for the system to have regularly and/orstrongly decaying solutions. The results here presented complete the onesin [10].MSC 2000. 34B15, 34C11.Keywords. p-laplacian operator, regularly decaying solutions, stronglydecaying solutions, fixed point theorems.1 IntroductionIn this contribution we present some asymptotic results for a system of two couplednonlinear equations with the 1-dimensional p-laplacian operator Ψp(u) = |u|p−2u,u ∈ R, of the form[r(t)Ψp(x′)]′ = −F (t, x, y)[q(t)Ψk(y′)]′ = G(t, x, y)(S) Supported by the Grants No. 201/01/P041 and No. 201/01/0079 of the Czech GrantAgency and by Italian C.N.R.This is in the final form and completes recent results by the authors.280 M. Marini, S. Matucci and P. Řehákwhere we assume p, k > 1 and(i) r, q : [0,∞) -→ (0,∞) continuous,(ii) F, G : [0,∞) × (0, δ] × (0, δ] -→ (0,∞) continuous, with δ a suitably smallpositive constant,(iii) ∃f, g : [0,∞)× (0, δ] -→ (0,∞) continuous s.t. F (t, u, v) ≤ f(t, v), G(t, u, v) ≤g(t, u), ∀(t, u, v) ∈ [0,∞)× (0, δ]× (0, δ].Systems of type (1) comes out in the study of the existence of radial solutions fornonlinear coupled elliptic systems with p-laplacian ∆pu = div(|∇u|p−2∇u)∆pu = F (|x|, u, v)∆kv = G(|x|, u, v)in exterior domains Ea = {x ∈ RN : |x| ≥ a}, N ≥ 2, p > 1, k > 1. Systems ofpartial differential equations of this form have been object of increasing interestin the last years, due to their relevance in applied sciences, especially in plasmaphysics, biomathematics, chemistry and, in general, in reaction-diffusion problems.Some relevant examples can be found in [2], [4], [5], [6], [7] and in the book [3] towhich we address the interested reader.The particular case of an ordinary differential system of two coupled generalizedEmden-Fowler equations of the form(p(t)|x′|α−1x′)′ = ϕ(t)yλ(q(t)|y′|β−1y′)′ = ψ(t)xµ.(2)has been recently studied in [8], [9], [12], with regard to existence of positive de-creasing solutions, but the sign condition on the nonlinear terms is opposite to ourcase. The results here presented are therefore both more general and complemen-tary to the corresponding ones in [8], [9],[12]. As both the nonlinear terms in (2)have positive sign, the dynamics which is described is completely different fromour case; further (2) seems to be more close to the case of a single equation[r(t)Ψp(x′)]′ = f(t, x).that has been considered, for instance, in [1], [11] (see also the references therein).Here we want to present some existence results which complete the ones ob-tained by the authors in [10]; in particular system (1) is more general than the oneconsidered in [10] since the nonlinear terms F and G in (1) depend on both theunknowns x and y. No assumption is done on the continuity and on the bounded-ness of F and G in a neighborhood of v = 0 and of u = 0 respectively, thus we cantreat the regular case and the singular case at the same time, and the forced caseas well. For sake of completeness we will also quote the necessary and sufficientconditions that can be derived for a Emden-Fowler type system, whose proofs canbe found in [10].We end this section stating some definitions.Zero Convergent Solutions of p-Laplacian Systems 281In this contribution we deal with the existence of decaying solutions of (1), i.e.solutions (x, y) such that x and y are eventually positive nonincreasing and x(∞) =y(∞) = 0. If (x, y) is a decaying solution of (1), then the first quasiderivative of x,x[1] := r(t)Ψp(x′(t)) is eventually negative decreasing and the first quasiderivativeof y, y[1] := q(t)Ψk(y′(t)) is eventually negative increasing. Thus they admit limitas t → ∞ and −∞ ≤ x[1](∞) < 0, −∞ < y[1](∞) ≤ 0. A decaying solution(x, y) is called regularly decaying if x[1] is bounded and y[1] tends to a (negative)nonzero limit, strongly decaying if x[1] is bounded and y[1] tends to zero. Theset of regularly decaying solutions will be denoted by DR and the set of stronglydecaying solutions with DS . In the next section we will find minimal conditions inorder that (1) has solutions in the class DR and in DS , using both a topologicalapproach (the Shauder-Tychonoff fixed point theorem in a Frechét space) andintegral inequalities. This method allows us also to obtain an asymptotic estimateof the convergence rate of the solutions.Finally, we close this section by giving two necessary conditions that can beeasily proved (see [10]); the first one will be always assumed in the following, thesecond one will be assumed only in the existence results in DR:– If (1) has at least one decaying solution then the condition∫ ∞0Ψp∗(1r(t))dt <∞ (H1)is satisfied, where p∗ is the conjugate numbers of p, i.e. 1/p+ 1/p∗ = 1.– If (1) has at least one regularly decaying solution then conditions (3) and∫ ∞0Ψk∗(1q(t))dt <∞ (H2)are satisfied, where k∗ is the conjugate numbers of k.2 Existence results in DR and in DS for (1)Concerning the existence of solutions in the class DR, the following result holds.Theorem 1. Assume (3), (4) and1. there exist a positive continuous functions ϕ1 on [0,∞), a nonnegative con-tinuous function ϕ2 on [0,∞) and a monotone positive continuous function f̂on (0, δ] such that for (t, u) ∈ [0,∞)× (0, δ]f(t, u) ≤ ϕ1(t)f̂(u) + ϕ2(t),∫ ∞0ϕ2(t) dt <∞,∫ ∞0ϕ1(t)f̂(∫ ∞tΨk∗(1q(τ))dτ)dt <∞;(5)282 M. Marini, S. Matucci and P. Řehák2. there exist a positive continuous functions γ1 on [0,∞), a nonnegative con-tinuous functions γ2 on [0,∞) and a monotone positive continuous functionĝ on (0, δ] such that for (t, u) ∈ [0,∞)× (0, δ]g(t, u) ≤ γ1(t)ĝ(u) + γ2(t),∫ ∞0γ2(t) dt <∞,∫ ∞0γ1(t)ĝ(∫ ∞tΨp∗(1r(τ))dτ)dt <∞.(6)Then (1) has solutions in the class DR.Proof. The argument is a slight generalization of the one employed in [10] in theproof of Theorem 1, in which the particular case of a nonforced system is consid-ered. We denote by C[t0,∞) the locally convex space of all continuous functionsdefined on [t0,∞) with the topology of uniform convergence on any compact subin-tervals of [t0,∞). Thus C[t0,∞) is a Fréchet space. We choose t0 ≥ 0 such that∫ ∞t0ϕ2(t) dt ≤14,∫ ∞t0ϕ1(t)f̂(∫ ∞tΨk∗(1q(τ))dτ)dt ≤ 14,∫ ∞t0γ2(t) dt ≤14,∫ ∞t0γ1(t)ĝ(∫ ∞tΨp∗(1r(τ))dτ)dt ≤ 14.(7)Letm = max{∫ ∞t0Ψk∗(1q(τ))dτ ;∫ ∞t0Ψp∗(1r(τ))dτ}and, without loss of generality, assume 2m ≤ δ.Consider the set Ω ⊂ C[t0,∞)× C[t0,∞) given byΩ ={(u, v) ∈ C[t0,∞)× C[t0,∞) such that:∫ ∞tΨp∗(M1r(τ))dτ ≤ u(t) ≤∫ ∞tΨp∗(M2r(τ))dτ,∫ ∞tΨk∗(N1q(τ))dτ ≤ v(t) ≤∫ ∞tΨk∗(N2q(τ))dτ}where Mi, Ni, i = 1, 2 are four suitable positive constants. On this set we definethe operator T with values in C[t0,∞)× C[t0,∞), byT (u, v) = (T1(u, v), T2(u, v))T1(u, v)(t) =∫ ∞tΨp∗[1r(s)(M2 −∫ ∞sF (τ, u(τ), v(τ)) dτ)]ds,T2(u, v)(t) =∫ ∞tΨk∗[1q(s)(N1 +∫ ∞sG(τ, u(τ), v(τ)) dτ)]ds.We have to prove that T is continuous and maps Ω into a compact subset of Ω.Zero Convergent Solutions of p-Laplacian Systems 283First we consider the case f̂ and ĝ nondecreasing on (0, δ]; for this case letM1 = N1 = 1/2, M2 = N2 = 1.(i) T (Ω) ⊂ Ω. The positivity of F and G immediately implies, for every (u, v) ∈ ΩT1(u, v)(t) ≤∫ ∞tΨp∗(1r(s))ds, T2(u, v)(t) ≥∫ ∞tΨk∗(1/2q(s))ds.The assumptions on F and G, together with (5) and (7), imply that for any(u, v) ∈ Ω it holds∫ ∞t0F (t, u(t), v(t)) dt ≤∫ ∞t0ϕ1(t)f̂(∫ ∞tΨk∗(1q(τ))dτ)dt+∫ ∞t0ϕ2(t) dt ≤12,∫ ∞t0G(t, u(t), v(t)) dt ≤∫ ∞t0γ1(t)ĝ(∫ ∞tΨp∗(1r(τ))dτ)dt+∫ ∞t0γ2(t) dt ≤12.These inequalities implyT1(u, v)(t) ≥∫ ∞tΨp∗(1/2r(s))ds, T2(u, v)(t) ≤∫ ∞tΨk∗(1q(s))ds,and (i) follows.(ii) T (Ω) is relatively compact. Since T (Ω) ⊂ Ω, functions in T (Ω) are equi-bounded. The equicontinuity of functions in T (Ω) easily follows by observing that,in virtue of the above estimates, for any (u, v) ∈ Ω it holds0 ≤ −(T1(u, v))′(t) ≤ Ψp∗(1r(t)), 0 ≤ −(T2(u, v))′(t) ≤ Ψk∗(1q(t)).(iii) T is continuous in Ω ⊂ C[t0,∞) × C[t0,∞). Let {(un, vn)}, n ∈ N, be asequence in Ω which uniformly converges on every compact interval I of [t0,∞)to (ū, v̄) ∈ Ω. In view of the assumption on F and of the upper bound (5), theLebesgue dominated convergence theorem and the uniform convergence on I ofthe sequence {F (t, un, vn(t))} imply that the sequence{∫∞t F (τ, un(τ), vn(τ)) dτ}uniformly converges to∫∞t F (τ, ū(τ), v̄(τ)) dτ on I. Analogously, the upper bound0 ≤ Ψp∗[1r(t)(1−∫ ∞tF (τ, un(τ), vn(τ)) dτ)]≤ Ψp∗(1r(t))allows us to apply again the Lebesgue dominated convergence theorem to thesequence {Ψp∗[1r(t)(1−∫ ∞tF (τ, un(τ), vn(τ)) dτ)]}.It follows that the sequence {T1(un, vn)} uniformly converges on I to T1(ū, v̄), thatis the continuity of T1. The argument for the continuity of T2 is quite the sameand (iii) is proved.284 M. Marini, S. Matucci and P. ŘehákSince (i), (ii), (iii) are satisfied, the Schauder-Tychonoff theorem implies thatthe operator T has a fixed point (z, w) ∈ Ω. It is easy to show that (z, w) issolution of (1) on [t0,∞) and (z, w) ∈ DR.In the case f̂ and ĝ nonincreasing on (0, δ] the assertion can be proved byusing an argument analogous to that given in the first part of the proof andchoosing M1 = N1 = 1, M2 = N2 = 3/2. Finally, when f̂ is nondecreasing and ĝnonincreasing on (0, δ] or vice-versa, it is sufficient to choose M1 = N2 = 1, M2 =3/2, N1 = 1/2 or N1 =M2 = 1, M1 = 1/2, N2 = 3/2 respectively. The details areleft to the reader. @AOther existence results in the class DR can be found in [10] (Th. 1 and Prop. 1)in case that the nonlinearities in (1) satisfy some additional assumptions.Remark 2. The assumptions in Theorem 1 are quite general and Theorem 1 canbe applied also to systems with singular forcing terms; an interesting case is whenF (t, u, v) = ϕ(t)uλ1v−λ2 and/or G(t, u, v) = ψ(t)u−µ1vµ2 , with λ1, µ2 ≥ 0 andλ2, µ1 > 0.With regards to existence of solutions of (1) in the class DS , here we treat onlythe case in which F (t, u, v) and G(t, u, v) are both singular in a neighborhood ofv = 0 and u = 0 respectively. The remaining cases, i.e. F and G both regularfunctions or one regular and the other singular in a neighborhood of v = 0 andu = 0 respectively, can be obtained with minor changes from the results in [10](Th. 2, Th. 3, Prop. 2). The details are left to the reader. To state the followingexistence, we need a further assumption in addition to (i)-(iii).Theorem 3. Assume(iv) ∃h : [0,∞) × (0, δ] -→ (0,∞) continuous s.t. G(t, u, v) ≥ h(t, u), ∀(t, u, v) ∈[0,∞)× (0, δ]× (0, δ].Let (3) be satisfied and suppose that there exist two positive continuous functionsγ1, ϕ1 on [0,∞), two nonnegative continuous functions γ2, ϕ2 on [0,∞), threenondecreasing positive continuous functions ĥ, ĝ, f̂ on (0, δ] and two positive con-stants 0 < N < M , such that for (t, u) ∈ [0,∞)× (0, δ]:h(t, u) ≥ γ1(t)ĥ(u) + γ2(t), g(t, u) ≤ γ1(t)ĝ(u) + γ2(t),f(t, u) ≤ ϕ1(t)f̂(u) + ϕ2(t)∫ ∞0γ2(t) dt <∞,∫ ∞0ϕ2(t) dt <∞,∫ ∞0Ψk∗(DN (t)q(t))dt <∞,∫ ∞0ϕ1(t)f̂(∫ ∞tΨk∗(CM (s)q(s))ds)dt <∞,whereDN (t) =∫ ∞t[γ1(s)ĝ(∫ ∞sΨp∗(Nr(τ))dτ)+ γ2(s)]dsCM (t) =∫ ∞t[γ1(s)ĥ(∫ ∞sΨp∗(Mr(τ))dτ)+ γ2(s)]ds.Zero Convergent Solutions of p-Laplacian Systems 285Then (1) has solutions in the class DS.Proof. Choose t0 ≥ 0 such that∫ ∞t0[ϕ1(t)f̂(∫ ∞tΨk∗(CM (s)q(s))ds)+ ϕ2(t)]dt ≤M −N.Letm1 = max{∫ ∞t0Ψp∗(Mr(t))dt,∫ ∞t0Ψk∗(DNq(t))dt}and, without loss of generality, suppose 2m1 < δ. Define the set Ω as followsΩ ={(u, v) ∈ C[t0,∞)× C[t0,∞) such that:∫ ∞tΨp∗(Nr(τ))dτ ≤ u(t) ≤∫ ∞tΨp∗(Mr(τ))dτ,∫ ∞tΨk∗(CM (σ)q(σ))dσ ≤ v(t) ≤∫ ∞tΨk∗(DN(σ)q(σ))dσ}Consider the operator T̂ : Ω -→ C[t0,∞)×C[t0,∞), T̃ (u, v) = (T̃1(u, v), T̃2(u, v)),given byT̃1(u, v)(t) =∫ ∞tΨp∗(1r(s)(M −∫ ∞sF (τ, u(τ), v(τ)) dτ))ds,T̃2(u, v)(t) =∫ ∞tΨk∗(1q(s)∫ ∞sG(τ, u(τ), v(τ)) dτ)ds.Clearly T̃1(u, v)(t) ≤∫∞t Ψp∗(Mr(τ))dτ for every (u, v) ∈ Ω, t ≥ t0. The followinginequalities hold for s ≥ t0:∫ ∞sF (t, u(t), v(t))dt ≤∫ ∞sf(t, v(t))dt ≤∫ ∞s(ϕ1(t)f̂(v(t)) + ϕ2(t))dt≤∫ ∞t0[ϕ1(t)f̂(∫ ∞tΨk∗(CM (s)q(s))ds)+ ϕ2(t)]dt ≤M −N ;∫ ∞sG(t, u(t), v(t))dt ≤∫ ∞sg(t, u(t))dt ≤∫ ∞s(γ1(t)ĝ(u(t)) + γ2(t))dt≤∫ ∞s[γ1(t)ĝ(∫ ∞tΨp∗(Nr(s))ds)+ γ2(t)]dt = DN (s);∫ ∞sG(t, u(t), v(t))dt ≥∫ ∞sh(t, u(t))dt ≥∫ ∞s(γ1(t)ĥ(u(t)) + γ2(t))dt≥∫ ∞s[γ1(t)ĥ(∫ ∞tΨp∗(Mr(s))ds)+ γ2(t)]dt = CM (s).From the above estimates it follows that the operator T̃ maps Ω into itself. Toapply Tychonov fixed point theorem, it is sufficient to show that T̃ (Ω) is relatively286 M. Marini, S. Matucci and P. Řehákcompact and T̃ is continuous in Ω. The argument is similar to that given in theproof of Theorem 1 and the details are left to the reader. Then the operator T̃has at least one fixed point (z, w). It is easy to show that (z, w) is solution of (1)on [t0,∞) and both components of the fixed point are positive in each interval of[t0,∞): so (z, w) ∈ DS . @ARemark 4. Unlike Theorem 1, that can be applied to both the regular case and thesingular one (see Remark 2), Theorem 3 is for the singular case only. Existenceresults for the regular case can be found in [10] and applied to (1) with minorchanges. This fact depends on the structure of the operator T̃ whose fixed pointsare singular solutions of (1).The conditions stated in Theorem 1 and in Theorem 3 are sharp conditions forexistence of solutions of (1) in DR and in DS respectively, since these conditionsbecomes also necessary in case of a forced Emden-Fowler type system[r(t)Ψp(x′)]′ = −ϕ1(t)Ψµ(y)− ϕ2(t), µ = 1[q(t)Ψk(y′)]′ = γ1(t)Ψν(x) + γ2(t), ν = 1,(8)where we assume p, k > 1, µ, ν = 1, ϕ1, γ1 : [0,∞) -→ (0,∞) continuous, ϕ2, γ2 :[0,∞) -→ [0,∞) continuous, and ϕ2, γ2 ∈ L1(0,∞). The following result holds,whose proof can be found in [10]Theorem 5 ([10] Th. 3, Th. 4, Prop. 3).• The Emden-Fowler forced system (8) has solutions in the class DR if and onlyif conditions (3), (4) and∫ ∞0ϕ1(t)(∫ ∞tΨk∗(1q(τ))dτ)µ−1dt <∞∫ ∞0γ1(t)(∫ ∞tΨp∗(1r(τ))dτ)ν−1dt <∞are satisfied.• The Emden-Fowler forced system (8) has solutions in the class DS if and onlyif conditions (3) and∫ ∞0ϕ1(t)(∫ ∞tϑ(s) ds)µ−1dt <∞are satisfied, whereϑ(t) = Ψk∗[1q(t)∫ ∞t(γ1(s)(∫ ∞sΨp∗(1r(τ))dτ)ν−1+ γ2(s))ds].Comments and examples that illustrate the previous result can be found in [10].Zero Convergent Solutions of p-Laplacian Systems 287References1. M. Cecchi, Z. Dosla, M. Marini, On nonoscillatory solutions of differential equationswith p-Laplacian, Advances in Math. Sci. and Appl. 11 1 (2001), 419-436.2. P. Clémen, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solu-tions for a nonvariational quasilinear elliptic system, J. Diff. Eq. 166 (2000), 455–477.3. J.I. Diaz, Nonlinear Partial Differential Equations And Free Boundaries, Volume I:Elliptic Equations, Research Notes in Mathematics 106, Pitman Publishing Ltd.,London, 1985.4. J. Fleckinger, R. Manásevich and F. de Thélin, Global bifurcation for the first eigen-value for a system of p-laplacians, Math. Nachr. 182 (1996), 217–242.5. J. Fleckinger, R. Pardo and F. de Thélin, Four parameter bifurcation for a p-Laplacian system, Elect. J. Diff. Eq. 2001 (2001), 1–15.6. Y.X. Huang and J.W.H. So, On bifurcation and existence of positive solutions for acertain p-Laplacian system, Rocky Mount. J. Math. 25 (1995), 285–297.7. D. Jiang and H. Liu, On the existence of nonnegative radial solutions for p-Laplacianelliptic systems, Ann. Polonici Math. LXXLI (1999), 19–29.8. T. Kusano and T. Tanigawa, Existence of positive decreasing solutions to a class ofsecond-order nonlinear differential systems, Fukuoka Univ. Sci. Rep. 29 2 (1999),231–242.9. T. Kusano and T. Tanigawa, Positive decreasing solutions of systems of second ordersingular differential equations, J. of Inequal. and Appl. 5 (2000), 581–602.10. M. Marini, S. Matucci and P. Řehák, On decay of coupled nonlinear differentialsystems, to appear in Adv. Math. Sci. Appl.11. S. Matucci, On asymptotic decaying solutions for a class of second order differentialequations, Arch. Math.(Brno) 35 (1999), 275-284.12. T. Tanigawa, Positive decreasing solutions of systems of second order quasilineardifferential equations, Funk. Ekvac. 43 (2000), 361–380.

Zero convergent solutions for a class of $p$-Laplacian systems

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Zero convergent solutions for a class of $p$ -Laplacian systems