Ouscula Math. 38 no. 2 2018 187 199 htts://doi.org/10.7494/omath.2018.38.2.187 Ouscula Mathematica EXISTENCE RESULTS FOR KIRCHHOFF TYPE SYSTEMS WITH SINGULAR NONLINEARITY A. Firoujai G.A. Afroui and S. Talebi Communicated by Dušan Reovš Abstract. Using the method of sub-suer solutions we study the existence of ositive solutions for a class of singular nonlinear semiositone systems involving nonlocal oerator. Keywords: sub-suersolution infinite semiositone systems singular weights Kirchhoff-tye. Mathematics Subject Classification: 35J55 35J65. 1. INTRODUCTION We study the existence of ositive solutions to the singular infinite semiositone system M 1 x α u dx div x α u 2 u = x α+1+β a 1 u 1 f 1 u b1 v x γ M 2 x α v dx div x α v 2 v 1.1 = x α+1+β a 2 v 1 f 2 v b2 u x γ u = v = 0 x where is a bounded smooth domain of R N N 3 with 0 1 < < N 0 α < N γ 0 1 and a 1 a 2 b 1 b 2 β are ositive constants and f i : 0 R i = 1 2 are continuous functions and M i : 0 R + i = 1 2 aside from being continuous and nondecreasing functions and 0 < M i0 M i t M i for all t 0 verify: H There exist t 2 > t 1 > 0 such that Mit2 2 N 2 t2 > Mit1 2 N 2 t1 see 10. c Wydawnictwa AGH Krakow 2018 187
188 A. Firoujai G.A. Afroui and S. Talebi A tyical examle of a function satisfying this condition is M i t = M i0 + at i = 1 2 with a 0 and for all t 0. We make the following assumtions: A1 There exist L > 0 and b > 1 such that f i u Lu b for all u 0 and i = 1 2. A2 There exists a constant S > 0 such that a i u 1 f i u < S for u 0 and i = 1 2. A simle examle of f i satisfying these assumtions is f i u = u b i = 1 2 for any b > 1. System 1.1 is related to the stationary roblem of a model introduced by Kirchhoff 12. More recisely Kirchhoff roosed a model given by the equation ρ 2 u t 2 P0 h + E 2L L 0 u 2 2 u dx = 0 1.2 x x2 where ρ P 0 h E are all constants. This equation extends the classical d Alembert wave equation. A distinguishing feature of equation 1.2 is that the equation has a nonlocal coefficient P0 h + E L 2L 0 u x 2 dx which deends on the average 1 L 2L 0 u x 2 dx. Hence the equation is no longer a ointwise identity. We refer to 19 for additional result on Kirchhoff equations. In recent years there has been considerable rogress on the study of nonlocal roblems see 15 17 18. Nonlocal roblems can be used for modeling for examle hysical and biological systems for which u describes a rocess which deends on the average of itself such as the oulation density. On the other hand ellitic roblems involving more general oerator such as the degenerate quasilinear ellitic oerator given by div x α u 2 u were motivated by the following Caffarelli Kohn and Nirenberg s inequality see 4 16 22. The study of this tye of roblem is motivated by its various alications for examle in fluid mechanics in newtonian fluids in flow through orous media and in glaciology see 3 7. So the study of ositive solutions of singular ellitic roblems has more ractical meanings. Let and Then F h k = a 1 h 1 f 1 h b 1 k γ Gh k = a 2 k q 1 f 2 k b 2 h γ. lim F h k = lim Gh k = hk 00 hk 00 and hence we refer to 1.1 as an infinite semiositone system. In 13 the authors discussed the single roblem 1.1 when M 1 t 1 α = 0 = β = 2 and see 20 for the single equation case when M 1 t 1. Here we focus on further extending the study in 13 20 for infinities semiositone Kirchhoff tye systems involving singularity. Our aroach is based on the method of sub-suersolutions see 5 8.
Existence results for Kirchhoff tye systems with singular nonlinearity 189 2. PRELIMINARIES AND EXISTING RESULT In this aer we denote W 1 0 x α the comletion of C 0 with resect to the norm u = x α u dx 1. To recisely state our existence result we consider the eigenvalue roblem { div x α φ 2 φ = λ x α+1+β φ 2 φ x φ = 0 x. 2.1 Let φ 1 be the eigenfunction corresonding to the first eigenvalue λ 1 of 2.1 such that φ 1 x > 0 in and φ 1 = 1 see 14 21. It can be shown that φ 1 n < 0 on. Here n is the outward normal. We will also consider the unique solution ζ x W 1 0 x α for the roblem { div x α ζ 2 ζ = x α+1+β x ζ = 0 x to discuss our existence result. It is well known that ζ x > 0 in and ζx n < 0 on see 14. A air of nonnegative functions ψ 1 ψ 2 1 2 are called a sub-solution and suer-solution of 1.1 if they satisfy ψ 1 ψ 2 = 0 0 = 1 2 on and M 1 x α ψ 1 dx x α ψ 1 2 ψ 1 x α+1+β a 1 ψ 1 1 f 1 ψ 1 b1 ψ γ 2 M 2 x α ψ 2 dx x α ψ 2 2 ψ 2 x α+1+β a 2 ψ 1 2 f 2 ψ 2 b2 ψ γ 1 M 1 x α 1 dx x α 1 2 1 x α+1+β a 1 1 1 f 1 1 b1 γ 2 M 2 x α 2 dx x α 2 2 2 x α+1+β a 2 1 2 f 2 2 b2 γ 1 for all w W = {w C 0 w 0 x }. A key role in our arguments will be layed by the following auxiliary result. Its roof is similar to those resented in 6. The reader can consult further the aers 1 2 11.
190 A. Firoujai G.A. Afroui and S. Talebi Lemma 2.1. Assume that M : R + 0 R+ is continuous and increasing and there exists m 0 > 0 such that Mt m 0 for all t R + 1 0. If the functions u v W0 x α satisfy M x α u dx x α u 2 u ϕdx M x α v dx x α v 2 v ϕdx for all ϕ W 1 0 x α ϕ 0 then u v in. From Lemma 2.1 we can establish the basic rincile of the sub-and suersolution method for nonlocal systems. Indeed we consider the following nonlocal system M 1 x α u dx div x α u 2 u = x α+1+β hx u v x M 2 x α v dx div x α v 2 v = x β+1+β kx u v x u = v = 0 x 2.2 where is a bounded smooth domain of R N and h k : R R R satisfy the following conditions: HK1 hx s t and kx s t are caratheodory functions and they are bounded if s t belong to bounded sets. HK2 There exists a function g : R R being continuous nondecreasing with g0 = 0 0 gs c1 + s min{q} for some c > 0 and alications s hx s t + gs and t kx s t + gt are nondecreasing for a.e x. If u v L with ux vx for a.e x we denote by u v the set {w L : ux wx vx for a.e x }. Using Lemma 2.1 and the method as in the roof of Theorem 2.4 of 14 see also Section 4 of 5 we can establish a version of the abstract lower and uer-solution method for our class of the oerators as follows. Proosition 2.2. Let M i : R + 0 R+ i = 1 2 are two continuous and increasing functions 0 < M i M i t M i for all t R +. Assume that the functions h k satisfy the conditions HK 1 and HK 2. Assume that u v u v are resectively a weak subsolution and a weak suersolution of system 2.2 with ux ux and vx vx for a.e x. Then there exist a minimal u v and resectively a maximal u v weak solution for system 2.2 in the set u u v v. In articular every weak solution u v u u v v of system 2.2 satisfies u x ux u x and v x vx v x for a.e x.
Existence results for Kirchhoff tye systems with singular nonlinearity 191 Theorem 2.3. Assume that 1λ1 min{a 1 a 2 } > KK 1 + γ where K = max{m 1 M 2 } then there exists c > 0 such that if max{b 1 b 2 } c then the system 1.1 admits a ositive solution. Proof. We start with the construction of a ositive subsolution for 1.1. To get a ositive subsolution we can aly an anti-maximum rincile see 9 from which we know that there exist a δ 1 > 0 and a solution λ of { div x α 2 = x α+1+β λ 1 1 x = 0 x for λ λ 1 λ 1 + δ 1. Fix { 1 + γ } 1a1 λ 1 min λ 1 + δ 1. Let θ =. It is well known that > 0 in and n < 0 on where n is the outer unit normal to. Hence there exist ositive constants ɛ δ σ such that x α ɛ x δ 2.3 σ x 0 = \ δ where δ = {x dx δ}. Choose η 1 η 2 > 0 such that η 1 min x α+1+β and η 2 max x α+1+β in δ. We construct a subsolution ψ 1 ψ 2 of 1.1 using. Define 1 + γ 1 + γ where ψ 1 ψ 2 = M { M1 M = min Lθ M2 1 L M bθ 1 γ 1 b bθ 1 γ 1 Lθ b θ 1 1 L θ 1 1 b +1 1 b +1 1a1 M 1 bθ b 1a2 M 2 bθ b 1 b +1 1 b +1 }.
192 A. Firoujai G.A. Afroui and S. Talebi Let w W. Then a calculation shows that ψ 1 = M 1 γ M 1 x α ψ 1 dx M 1 M 1 = M 1 M 1 = M 1 M 1 = M 1 x α ψ 1 2 ψ 1 1 γ 1 x α 2 x α 2 x α+1+β 1 γ 1 1 γ 1 1 α 1 γ 1 x γ 1 + γ x α+1+β M 1 1 x α 1 1 γ 1 M γ 1 + γ 1 w 1 γ 1 x α+1+β M 1 1 γ 1 w dx and x α+1+β a 1 ψ 1 1 f 1 ψ 1 b 1 ψ γ 2 = x α+1+β a 1 M 1 1 + γ 1 + γ x α+1+β f 1 M x α+1+β b 1 M γ γ γ 1 1.
Existence results for Kirchhoff tye systems with singular nonlinearity 193 Similarly and M 2 M 2 x α ψ 2 dx x α+1+β M 1 1 x α ψ 2 2 ψ 2 x α+1+β M 1 1 γ 1 x α 1 1 γ 1 M 1 + γ x α+1+β a 2 ψ 1 2 f 2 ψ 2 b 2 = ψ γ 1 x α+1+β a 2 M 1 1 + γ 1 + γ x α+1+β f 2 M x α+1+β b 2 M γ γ γ γ 1 1. Let { 1 γ 1 c = min M 1 M 1 + γ 1 γ 1 M 2 M 1 + γ M 1 + γ 1 + γ M 1 + γ γ ɛ η 2 1 + γ γ ɛ η 2 γσ 1 + γ γσ 1 + γ 1a1 M 1 1a2 } M 2. First we consider the case when x δ. We have x α ɛ on δ. Since we have x α+1+β M 1 M 1 1 M 1 1 a1 1 + γ x α+1+β a 1 M 1 1 + γ 1 1 2.4
194 A. Firoujai G.A. Afroui and S. Talebi and from the choice of M we know that LM b +1 θ b bθ 1 γ 1 M1. 2.5 1 + γ By 2.5 and A 1 we have x α+1+β M 1 M 1 1 γ 1 x α+1+β LM b 1 + γ 1 + γ x α+1+β f 1 M b b. 2.6 Next from 2.3 and definition of c we have x α M 1 M 1 1 γ 1 1 + γ x α+1+β b 1 M γ γ and x α 1 1 γ 1 M 1 M γ x α+1+β b 1 1 + γ M γ γ γ. 2.7 Hence by using 2.4 2.6 and 2.7 for b 1 c we have M 1 δ x α ψ 1 dx x α ψ 1 2 ψ 1 δ = δ δ x α+1+β a 1 M 1 1 + γ 1 + γ x α+1+β f 1 M x α+1+β b 1 M γ γ γ 1 1 1 α+1+β x a 1 ψ 1 1 f 1 ψ 1 b 1 ψ γ 2. 2.8
Existence results for Kirchhoff tye systems with singular nonlinearity 195 Similarly M 2 δ x α ψ 2 dx x α ψ 2 2 ψ 2 δ = δ δ x α+1+β a 2 M 1 1 + γ 1 + γ x α+1+β f 2 M x α+1+β b 2 M γ γ γ 1 1 α+1+β x a 2 ψ 1 2 f 2 ψ 2 b 2 ψ γ 1. 2.9 On the other hand on 0 = \ δ we have σ and from the definition of c for b 1 c we have b 1 M γ γ 1 M 1 σ 1 + γ 1a1 M 1 1 M 1 1 + γ 1a1 M 1. 2.10 Also from the choice of M we have b b LM b +1 1 + γ 1 1 1 + γ 1a1 M 1. 2.11
196 A. Firoujai G.A. Afroui and S. Talebi Hence from 2.10 and 2.11 we have M 1 x α ψ 1 dx x α ψ 1 2 ψ 1 0 M 1 0 x α+1+β M 1 1 0 M 1 x α+1+β M 1 1 γ 1 x α 1 1 γ 1 M γ 1 + γ x α+1+β M 1 1 0 = M 1 x α+1+β 1 γ 0 x α+1+β 1 1 γ 0 = + M 1 1a1 0 0 1 + γ x α+1+β x α+1+β b 1 M γ 1 1 M 1 + 1 M 1 M 1 1 + γ LM b +1 1 + γ a 1 M 1 1 + γ b 1 γ M γ γ a 1 M 1 1 + γ γ γ = 0 1a1 b 1 b 1 + γ 1 1 1 1 1 f 1 M M γ b 1 LM b 1 + γ γ 1 + γ q b b x α+1+β a 1 ψ 1 1 f 1 ψ 1 b 1 ψ γ. 2 2.12
Existence results for Kirchhoff tye systems with singular nonlinearity 197 Similarly M 2 0 x α ψ 2 dx x α ψ 2 2 ψ 2 0 0 x α+1+β a 2 ψ 1 2 f 2 ψ 2 b 2 ψ γ 1. 2.13 By using 2.8 2.9 2.12 and 2.13 we see that ψ 1 ψ 2 is a sub-solution of 1.1. Next we construct a suer-solution 1 2 of 1.1 such that 1 2 ψ 1 ψ 2. Let S 1 1 S 1 1 1 2 = ζ x ζ x. M 10 M 20 By A 2 and choosing a large constant S we shall verify that 1 2 is a suer-solution of 1.1. To this end let w W. Then we have M 1 x α 1 dx x α 1 2 1 S x α+1+β Similarly M 2 x α+1+β a 1 1 1 f 1 1 b 1 γ 2 x α 2 dx. x α 2 2 2 x α+1+β a 2 1 2 f 2 2 b 2 γ 1. Thus 1 2 is a suer-solution of 1.1. Finally we can choose S 1 such that ψ 1 ψ 2 1 2 in. Hence if max{b 1 b 2 } c by Lemma 2.1 there exists a ositive solution u v of 1.1 such that ψ 1 ψ 2 u v 1 2. This comletes the roof of Theorem 2.3. REFERENCES 1 G.A. Afroui N.T. Chung S. Shakeri Existence of ositive solutions for Kirchhoff tye equations Electron. J. Differential Equations 180 2013 1 8. 2 G.A. Afroui N.T. Chung S. Shakeri Existence of ositive solutions for Kirchhoff tye systems with singular weights submitted.
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Existence results for Kirchhoff tye systems with singular nonlinearity 199 A. Firoujai Firoujai@hd.nu.ac.ir Pyame Noor University Faculty of Basic Sciences Deartment of Mathematics Tehran Iran G.A. Afroui afroui@um.ac.ir Deartment of Mathematics Faculty of Mathematical Sciences University of Maandaran Babolsar Iran S. Talebi talebi_s@yahoo.com Deartment of Mathematics Faculty of Basic Sciences Pyame Noor University Mashhad Iran Received: February 17 2017. Revised: July 16 2017. Acceted: August 22 2017.